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Palindromic gapful numbers

From Rosetta Code


Task
Palindromic gapful numbers
You are encouraged to solve this task according to the task description, using any language you may know.

Numbers   (positive integers expressed in base ten)   that are (evenly) divisible by the number formed by the first and last digit are known as   gapful numbers.


Evenly divisible   means divisible with   no   remainder.


All   one─   and two─digit   numbers have this property and are trivially excluded.   Only numbers   100   will be considered for this Rosetta Code task.


Example

1037   is a   gapful   number because it is evenly divisible by the number   17   which is formed by the first and last decimal digits of   1037.


A palindromic number is   (for this task, a positive integer expressed in base ten),   when the number is reversed,   is the same as the original number.


Task
  •   Show   (nine sets)   the first   20   palindromic gapful numbers that   end   with:
  •   the digit   1
  •   the digit   2
  •   the digit   3
  •   the digit   4
  •   the digit   5
  •   the digit   6
  •   the digit   7
  •   the digit   8
  •   the digit   9
  •   Show   (nine sets, like above)   of palindromic gapful numbers:
  •   the last   15   palindromic gapful numbers   (out of      100)
  •   the last   10   palindromic gapful numbers   (out of   1,000)       {optional}


For other ways of expressing the (above) requirements, see the   discussion   page.


Note

All palindromic gapful numbers are divisible by eleven.


Related tasks


Also see



AppleScript[edit]

on doTask()
set part1 to {"First 20 palindromic gapful numbers > 100 ending with each digit from 1 to 9:"}
set part2 to {"86th to 100th such:"}
set part3 to {"991st to 1000th:"}
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to " "
repeat with endDigit from 1 to 9
set {collector1, collector2, collector3} to {{}, {}, {}}
set outerNumber to endDigit * 11 -- Number formed from the palindromes' first and last digits.
set oddDigitCount to true -- Starting with palindromes in the hundreds.
set baseHi to endDigit * 10 -- Number formed from just the "high end" digits, initially endDigit and a middle 0.
set hi to baseHi
set carryCheck to hi + 10 -- Number reached when incrementing the "high end" number changes its first digit.
set inc to 10 -- Incrementor for the middle digit(s) of the palindromes themselves.
set counter to 0
set maxNeeded to 1000
set done to false
repeat until (done)
-- Work out every 10th palindrome (middle digit = 0) from the current "high end" number.
set pal to hi
if (oddDigitCount) then
set temp to hi div 10
else
set temp to hi
end if
repeat until (temp is 0)
set pal to pal * 10 + temp mod 10
set temp to temp div 10
end repeat
-- Check the result and the following 9 palindromes (derived by incrementing the middle digit(s))
-- and store as text any which are both gapful and the ones required.
repeat 10 times
if (pal mod outerNumber is 0) then
set counter to counter + 1
if (counter ≤ 20) then
set end of collector1 to intText(pal)
else if (counter < 86) then
else if (counter ≤ 100) then
set end of collector2 to intText(pal)
else if (counter < 991) then
else --if (counter ≤ 1000) then
set end of collector3 to intText(pal)
set done to (counter = maxNeeded)
if (done) then exit repeat
end if
end if
set pal to pal + inc
end repeat
-- Increment the high end number's penultimate digit after every 10th palindrome.
-- If a carry changes its first digit, reset for longer palindromes.
set hi to hi + 10
if (hi = carryCheck) then
set oddDigitCount to (not oddDigitCount)
if (oddDigitCount) then
set baseHi to baseHi * 10
set carryCheck to carryCheck * 10
set inc to inc div 11 * 10
else
set inc to inc * 11
end if
set hi to baseHi
end if
end repeat
set {end of part1, end of part2, end of part3} to {collector1 as text, collector2 as text, collector3 as text}
end repeat
set AppleScript's text item delimiters to linefeed
set output to {part1, "", part2, "", part3} as text
set AppleScript's text item delimiters to astid
 
return output
end doTask
 
on intText(n)
if (n < 100000000) then return n as text
set txt to text 2 thru end of ((100000000 + (n mod 100000000 as integer)) as text)
set n to n div 100000000
repeat
set lo to n mod 100000000 as integer
set n to n div 100000000
if (n is 0) then return (lo as text) & txt
set txt to (text 2 thru end of ((100000000 + lo) as text)) & txt
end repeat
end intText
 
return doTask()
Output:
"First 20 palindromic gapful numbers > 100 ending with each digit from 1 to 9:
121  1001  1111  1221  1331  1441  1551  1661  1771  1881  1991  10901  11011  12221  13431  14641  15851  17171  18381  19591
242  2002  2112  2222  2332  2442  2552  2662  2772  2882  2992  20702  21912  22022  23232  24442  25652  26862  28182  29392
363  3003  3333  3663  3993  31713  33033  36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333
484  4004  4224  4444  4664  4884  40304  42724  44044  46464  48884  400004  401104  402204  403304  404404  405504  406604  407704  408804
5005  5115  5225  5335  5445  5555  5665  5775  5885  5995  50105  51315  52525  53735  54945  55055  56265  57475  58685  59895
6006  6336  6666  6996  61116  64746  66066  69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336
7007  7777  77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087
8008  8448  8888  80608  86768  88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648
9009  9999  94149  99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999  9459549  9508059  9557559  9606069

86th to 100th such:
165561  166661  167761  168861  169961  170071  171171  172271  173371  174471  175571  176671  177771  178871  179971
265562  266662  267762  268862  269962  270072  271172  272272  273372  274472  275572  276672  277772  278872  279972
30366303  30399303  30422403  30455403  30488403  30511503  30544503  30577503  30600603  30633603  30666603  30699603  30722703  30755703  30788703
4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284
565565  566665  567765  568865  569965  570075  571175  572275  573375  574475  575575  576675  577775  578875  579975
60399306  60422406  60455406  60488406  60511506  60544506  60577506  60600606  60633606  60666606  60699606  60722706  60755706  60788706  60811806
72299227  72322327  72399327  72422427  72499427  72522527  72599527  72622627  72699627  72722727  72799727  72822827  72899827  72922927  72999927
80611608  80622608  80633608  80644608  80655608  80666608  80677608  80688608  80699608  80800808  80811808  80822808  80833808  80844808  80855808
95311359  95400459  95499459  95588559  95677659  95766759  95855859  95944959  96033069  96122169  96211269  96300369  96399369  96488469  96577569

991st to 1000th:
17799771  17800871  17811871  17822871  17833871  17844871  17855871  17866871  17877871  17888871
27799772  27800872  27811872  27822872  27833872  27844872  27855872  27866872  27877872  27888872
3084004803  3084334803  3084664803  3084994803  3085225803  3085555803  3085885803  3086116803  3086446803  3086776803
482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384
57800875  57811875  57822875  57833875  57844875  57855875  57866875  57877875  57888875  57899875
6084004806  6084334806  6084664806  6084994806  6085225806  6085555806  6085885806  6086116806  6086446806  6086776806
7452992547  7453223547  7453993547  7454224547  7454994547  7455225547  7455995547  7456226547  7456996547  7457227547
8085995808  8086006808  8086116808  8086226808  8086336808  8086446808  8086556808  8086666808  8086776808  8086886808
9675005769  9675995769  9676886769  9677777769  9678668769  9679559769  9680440869  9681331869  9682222869  9683113869"

Translation of Phix "Ludicrously fast …"[edit]

See the original's comments for an explanation of the logic. While this AppleScript version is developed directly from the original, comparing the two may be difficult as I've pulled things around a bit for AppleScript efficiency and have relabelled some of the variables for consistency and to try to understand the workings myself! For the set task, it's not actually as fast as the script above, which doesn't have the overhead of being able to handle ridiculously large numbers and which gets the three sets of results for each digit in a single sweep rather than starting at the beginning for each set. It's also not nearly as fast as the Phix original apparently is, taking about twelve seconds to complete the same tests. But that's still not bad. Like the original, it's constrained not by the resolution of the results it can return, but by how many of them it can count using native number classes.

use AppleScript version "2.4" -- Mac OS 10.10 (Yosemite) or later.
 
-- Return a script object containing the main handlers.
-- It could be loaded as a library instead if there were any point in having such a library.
on getWorks()
script theWorks
use framework "Foundation" -- For its NSMutableDictionary class and methods.
 
property outerDigit : missing value
property outerDigit_x11 : missing value
property searchLimit : missing value
property to_skip : missing value
property palcount : missing value
property skipd : {{}, {}, {}, {}, {}, {}, {}, {}, {}}
property skipdSub : missing value
property output : missing value
 
-- Work out what the remainder would be if the decimal number described as one or two
-- instances of the digit 'd' with 'gap' zeros between them and 'shift' zeros following,
-- and which may be too large for AppleScript, were divided by the value 'outerDigit_x11'.
on rmdr(d, gap, shift)
-- Unlike the original Phix, this handler doesn't maintain a large cache, but computes its
-- usable low dividend on the fly by reducing shifts > 9 to equivalents between 4 and 9.
set multiplier to 10 ^ ((shift - 4) mod 6 + 4)
if (gap > -1) then set multiplier to multiplier + 10 ^ ((gap + shift - 3) mod 6 + 4)
 
return (d * multiplier mod outerDigit_x11) as integer
end rmdr
 
-- Recursively find successive palindromic gapful numbers beginning and ending with
-- 'outerDigit' and keep text versions of those in the required count range.
on palindromicGapfuls(lhs, gap, remainder)
-- lhs: eg {9, 4, 5} of a potential 9459549 result. (Text eg. "945" in the original.)
-- gap: length of inner to be filled in
-- remainder: remainder previously derived from rmdr() results.
set shift to (count lhs) -- left shift (should always equal length(lhs), I think)
-- Here, 'skipd' isn't a monolithic dictionary with four-element keys, but a list with 9 sublists,
-- each containing 99 smaller dictionaries with two-element keys. The values of the original
-- 'outerDigit' and 'remainder' key elements are used to index this structure. Since 'outerDigit'
-- has but one value per outer digit, the 'outerDigit'th sublist is prefetched and assigned
-- to the property 'skipdSub' in the set-up for each sweep.
set |key| to {gap, shift}
set skip to (item (remainder + 1) of my skipdSub)'s objectForKey:(|key|)
if (skip is missing value) then -- Key not in dictionary. Set 'skip' high for the 'if' test below.
set skip to searchLimit
else
set skip to (skip as real) div 1
end if
if (palcount + skip > to_skip) then -- (Also if they're equal, originally.)
set skip to 0
set template to lhs & missing value & reverse of lhs -- Palindrome template list.
set mid to shift + 1 -- Index of its middle position.
set nextGap to gap - 2
repeat with d from 0 to 9
set nextRem to (remainder + rmdr(d, nextGap, shift)) mod outerDigit_x11
if (gap > 2) then
set skip to skip + palindromicGapfuls(lhs & d, nextGap, nextRem)
else if (nextRem is 0) then
set palcount to palcount + 1
if (palcount > to_skip) then
if (gap is 2) then
set item mid of template to {d, d} -- Not d*11 in case d is 0.
else -- if (gap is 1) then
set item mid of template to d
-- else if (gap is 0) then
-- set item mid of template to ""
-- else
-- set template to d
end if
set end of my output to template as text
else
set skip to skip + 1
end if
end if
if (palcount = searchLimit) then exit repeat
end repeat
if (palcount < to_skip) then (item (remainder + 1) of my skipdSub)'s setObject:(skip) forKey:(|key|)
else
set palcount to palcount + skip
end if
 
return skip
end palindromicGapfuls
 
-- Return a list of numeric texts representing the last 'keep' of the first 'searchLimit'
-- palindromic gapful numbers > 100 which begin and end with 'digit'.
on collect(digit, searchLimit, keep)
-- Initialise these script object properties for the current search.
set outerDigit to digit
set outerDigit_x11 to outerDigit * 11
set my searchLimit to searchLimit
set to_skip to searchLimit - keep
set palcount to 0
set skipdSub to item outerDigit of my skipd
set output to {}
-- Locals.
set gap to 1 -- Initially 1 digit between the outer pair.
set lhs to {outerDigit} -- Initial left end of palindomes.
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to ""
repeat until (palcount = searchLimit)
set remainder to rmdr(outerDigit, gap, 0)
palindromicGapfuls(lhs, gap, remainder)
set gap to gap + 1
end repeat
set AppleScript's text item delimiters to astid
 
return output
end collect
 
on initialiseSkipd()
repeat with thisList in my skipd
repeat 99 times
set end of thisList to current application's class "NSMutableDictionary"'s new()
end repeat
end repeat
end initialiseSkipd
end script
 
tell theWorks to initialiseSkipd()
return theWorks
end getWorks
 
on doTask()
set tests to {{20, 20, 1, 9}, {100, 15, 1, 9}, {1000, 10, 1, 9}, {10000, 5, 1, 9}, ¬
{100000, 1, 1, 9}, {1000000, 1, 1, 9}, {10000000, 1, 1, 9}, ¬
{100000000, 1, 9, 9}, {1.0E+9, 1, 9, 9}, {1.0E+10, 1, 9, 9}, ¬
{1.0E+11, 1, 9, 9}, {1.0E+12, 1, 9, 9}, ¬
{1.0E+13, 1, 9, 9}, {1.0E+14, 1, 9, 9}, ¬
{1.0E+15, 1, 2, 4}}
-- set tests to {{20, 20, 1, 9}, {100, 15, 1, 9}, {1000, 10, 1, 9}} -- The set task.
 
set output to {}
set theWorks to getWorks()
set astid to AppleScript's text item delimiters
set AppleScript's text item delimiters to " "
repeat with i from 1 to (count tests)
set {searchLimit, keep, firstEndDigit, lastEndDigit} to item i of tests
set |from| to searchLimit - keep + 1
if (searchLimit = keep) then
set r to "First " & searchLimit
else if (keep > 1) then
set r to "Last " & keep & " of first " & searchLimit
else
set r to (searchLimit as text) & "th"
end if
set s to " > 100 ending with "
if (keep > 1) then set s to "s" & s
set t to (lastEndDigit as text) & ":"
if (firstEndDigit is not lastEndDigit) then set t to "digits " & firstEndDigit & " to " & t
set end of output to ""
set end of output to r & " palindromic gapful number" & s & t
repeat with digit from firstEndDigit to lastEndDigit
set end of output to theWorks's (collect(digit, searchLimit, keep)) as text
end repeat
end repeat
set AppleScript's text item delimiters to linefeed
set output to (rest of output) as text
set AppleScript's text item delimiters to astid
 
return output
end doTask
 
doTask()
Output:
First 20 palindromic gapful numbers > 100 ending with digits 1 to 9:
121  1001  1111  1221  1331  1441  1551  1661  1771  1881  1991  10901  11011  12221  13431  14641  15851  17171  18381  19591
242  2002  2112  2222  2332  2442  2552  2662  2772  2882  2992  20702  21912  22022  23232  24442  25652  26862  28182  29392
363  3003  3333  3663  3993  31713  33033  36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333
484  4004  4224  4444  4664  4884  40304  42724  44044  46464  48884  400004  401104  402204  403304  404404  405504  406604  407704  408804
5005  5115  5225  5335  5445  5555  5665  5775  5885  5995  50105  51315  52525  53735  54945  55055  56265  57475  58685  59895
6006  6336  6666  6996  61116  64746  66066  69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336
7007  7777  77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087
8008  8448  8888  80608  86768  88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648
9009  9999  94149  99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999  9459549  9508059  9557559  9606069

Last 15 of first 100 palindromic gapful numbers > 100 ending with digits 1 to 9:
165561  166661  167761  168861  169961  170071  171171  172271  173371  174471  175571  176671  177771  178871  179971
265562  266662  267762  268862  269962  270072  271172  272272  273372  274472  275572  276672  277772  278872  279972
30366303  30399303  30422403  30455403  30488403  30511503  30544503  30577503  30600603  30633603  30666603  30699603  30722703  30755703  30788703
4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284
565565  566665  567765  568865  569965  570075  571175  572275  573375  574475  575575  576675  577775  578875  579975
60399306  60422406  60455406  60488406  60511506  60544506  60577506  60600606  60633606  60666606  60699606  60722706  60755706  60788706  60811806
72299227  72322327  72399327  72422427  72499427  72522527  72599527  72622627  72699627  72722727  72799727  72822827  72899827  72922927  72999927
80611608  80622608  80633608  80644608  80655608  80666608  80677608  80688608  80699608  80800808  80811808  80822808  80833808  80844808  80855808
95311359  95400459  95499459  95588559  95677659  95766759  95855859  95944959  96033069  96122169  96211269  96300369  96399369  96488469  96577569

Last 10 of first 1000 palindromic gapful numbers > 100 ending with digits 1 to 9:
17799771  17800871  17811871  17822871  17833871  17844871  17855871  17866871  17877871  17888871
27799772  27800872  27811872  27822872  27833872  27844872  27855872  27866872  27877872  27888872
3084004803  3084334803  3084664803  3084994803  3085225803  3085555803  3085885803  3086116803  3086446803  3086776803
482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384
57800875  57811875  57822875  57833875  57844875  57855875  57866875  57877875  57888875  57899875
6084004806  6084334806  6084664806  6084994806  6085225806  6085555806  6085885806  6086116806  6086446806  6086776806
7452992547  7453223547  7453993547  7454224547  7454994547  7455225547  7455995547  7456226547  7456996547  7457227547
8085995808  8086006808  8086116808  8086226808  8086336808  8086446808  8086556808  8086666808  8086776808  8086886808
9675005769  9675995769  9676886769  9677777769  9678668769  9679559769  9680440869  9681331869  9682222869  9683113869

Last 5 of first 10000 palindromic gapful numbers > 100 ending with digits 1 to 9:
1787557871  1787667871  1787777871  1787887871  1787997871
2787557872  2787667872  2787777872  2787887872  2787997872
308760067803  308763367803  308766667803  308769967803  308772277803
48327872384  48329192384  48330303384  48331513384  48332723384
5787557875  5787667875  5787777875  5787887875  5787997875
608763367806  608766667806  608769967806  608772277806  608775577806
746958859647  746961169647  746968869647  746971179647  746978879647
808691196808  808692296808  808693396808  808694496808  808695596808
968697796869  968706607869  968715517869  968724427869  968733337869

100000th palindromic gapful number > 100 ending with digits 1 to 9:
178788887871
278788887872
30878611687803
4833326233384
578789987875
60878611687806
74826144162847
80869688696808
96878077087869

1000000th palindromic gapful number > 100 ending with digits 1 to 9:
17878799787871
27878799787872
3087876666787803
483333272333384
57878799787875
6087876996787806
7487226666227847
8086969559696808
9687870990787869

10000000th palindromic gapful number > 100 ending with digits 1 to 9:
1787878888787871
2787878888787872
308787855558787803
48333332623333384
5787878998787875
608787855558787806
748867523325768847
808696968869696808
968787783387787869

100000000th palindromic gapful number > 100 ending with 9:
96878786855868787869

1.0E+9th palindromic gapful number > 100 ending with 9:
9687878775995778787869

1.0E+10th palindromic gapful number > 100 ending with 9:
968787878661166878787869

1.0E+11th palindromic gapful number > 100 ending with 9:
96878787877355377878787869

1.0E+12th palindromic gapful number > 100 ending with 9:
9687878787863773687878787869

1.0E+13th palindromic gapful number > 100 ending with 9:
968787878787711117787878787869

1.0E+14th palindromic gapful number > 100 ending with 9:
96878787878786133168787878787869

1.0E+15th palindromic gapful number > 100 ending with digits 2 to 4:
27878787878787888878787878787872
3087878787878783113878787878787803
483333333333333262333333333333384

C[edit]

Translation of: C++
#include <stdbool.h>
#include <stdio.h>
#include <stdint.h>
 
typedef uint64_t integer;
 
integer reverse(integer n) {
integer rev = 0;
while (n > 0) {
rev = rev * 10 + (n % 10);
n /= 10;
}
return rev;
}
 
typedef struct palgen_tag {
integer power;
integer next;
int digit;
bool even;
} palgen_t;
 
void init_palgen(palgen_t* palgen, int digit) {
palgen->power = 10;
palgen->next = digit * palgen->power - 1;
palgen->digit = digit;
palgen->even = false;
}
 
integer next_palindrome(palgen_t* p) {
++p->next;
if (p->next == p->power * (p->digit + 1)) {
if (p->even)
p->power *= 10;
p->next = p->digit * p->power;
p->even = !p->even;
}
return p->next * (p->even ? 10 * p->power : p->power)
+ reverse(p->even ? p->next : p->next/10);
}
 
bool gapful(integer n) {
integer m = n;
while (m >= 10)
m /= 10;
return n % (n % 10 + 10 * m) == 0;
}
 
void print(int len, integer array[][len]) {
for (int digit = 1; digit < 10; ++digit) {
printf("%d: ", digit);
for (int i = 0; i < len; ++i)
printf(" %llu", array[digit - 1][i]);
printf("\n");
}
}
 
int main() {
const int n1 = 20, n2 = 15, n3 = 10;
const int m1 = 100, m2 = 1000;
 
integer pg1[9][n1];
integer pg2[9][n2];
integer pg3[9][n3];
 
for (int digit = 1; digit < 10; ++digit) {
palgen_t pgen;
init_palgen(&pgen, digit);
for (int i = 0; i < m2; ) {
integer n = next_palindrome(&pgen);
if (!gapful(n))
continue;
if (i < n1)
pg1[digit - 1][i] = n;
else if (i < m1 && i >= m1 - n2)
pg2[digit - 1][i - (m1 - n2)] = n;
else if (i >= m2 - n3)
pg3[digit - 1][i - (m2 - n3)] = n;
++i;
}
}
 
printf("First %d palindromic gapful numbers ending in:\n", n1);
print(n1, pg1);
 
printf("\nLast %d of first %d palindromic gapful numbers ending in:\n", n2, m1);
print(n2, pg2);
 
printf("\nLast %d of first %d palindromic gapful numbers ending in:\n", n3, m2);
print(n3, pg3);
 
return 0;
}
Output:
First 20 palindromic gapful numbers ending in:
1:  121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
2:  242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
3:  363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
4:  484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
5:  5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
6:  6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
7:  7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
8:  8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
9:  9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

Last 15 of first 100 palindromic gapful numbers ending in:
1:  165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
2:  265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
3:  30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4:  4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284
5:  565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
6:  60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7:  72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8:  80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9:  95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

Last 10 of first 1000 palindromic gapful numbers ending in:
1:  17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871
2:  27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872
3:  3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4:  482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384
5:  57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875
6:  6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7:  7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8:  8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9:  9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

C++[edit]

The idea of generating palindromes first then testing for gapfulness was borrowed from other solutions.

#include <iostream>
#include <cstdint>
 
typedef uint64_t integer;
 
integer reverse(integer n) {
integer rev = 0;
while (n > 0) {
rev = rev * 10 + (n % 10);
n /= 10;
}
return rev;
}
 
// generates base 10 palindromes greater than 100 starting
// with the specified digit
class palindrome_generator {
public:
palindrome_generator(int digit) : power_(10), next_(digit * power_ - 1),
digit_(digit), even_(false) {}
integer next_palindrome() {
++next_;
if (next_ == power_ * (digit_ + 1)) {
if (even_)
power_ *= 10;
next_ = digit_ * power_;
even_ = !even_;
}
return next_ * (even_ ? 10 * power_ : power_)
+ reverse(even_ ? next_ : next_/10);
}
private:
integer power_;
integer next_;
int digit_;
bool even_;
};
 
bool gapful(integer n) {
integer m = n;
while (m >= 10)
m /= 10;
return n % (n % 10 + 10 * m) == 0;
}
 
template<size_t len>
void print(integer (&array)[9][len]) {
for (int digit = 1; digit < 10; ++digit) {
std::cout << digit << ":";
for (int i = 0; i < len; ++i)
std::cout << ' ' << array[digit - 1][i];
std::cout << '\n';
}
}
 
int main() {
const int n1 = 20, n2 = 15, n3 = 10;
const int m1 = 100, m2 = 1000;
 
integer pg1[9][n1];
integer pg2[9][n2];
integer pg3[9][n3];
 
for (int digit = 1; digit < 10; ++digit) {
palindrome_generator pgen(digit);
for (int i = 0; i < m2; ) {
integer n = pgen.next_palindrome();
if (!gapful(n))
continue;
if (i < n1)
pg1[digit - 1][i] = n;
else if (i < m1 && i >= m1 - n2)
pg2[digit - 1][i - (m1 - n2)] = n;
else if (i >= m2 - n3)
pg3[digit - 1][i - (m2 - n3)] = n;
++i;
}
}
 
std::cout << "First " << n1 << " palindromic gapful numbers ending in:\n";
print(pg1);
 
std::cout << "\nLast " << n2 << " of first " << m1 << " palindromic gapful numbers ending in:\n";
print(pg2);
 
std::cout << "\nLast " << n3 << " of first " << m2 << " palindromic gapful numbers ending in:\n";
print(pg3);
 
return 0;
}
Output:
First 20 palindromic gapful numbers ending in:
1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

Last 15 of first 100 palindromic gapful numbers ending in:
1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284
5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

Last 10 of first 1000 palindromic gapful numbers ending in:
1: 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871
2: 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4: 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384
5: 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

Crystal[edit]

Brute force and slow[edit]

def palindromesgapful(digit, pow)
r1 = (10_u64**pow + 1) * digit
r2 = 10_u64**pow * (digit + 1)
nn = digit * 11
(r1...r2).select { |i| n = i.to_s; n == n.reverse && i.divisible_by?(nn) }
end
 
def digitscount(digit, count)
pow = 2
nums = [] of UInt64
while nums.size < count
nums += palindromesgapful(digit, pow)
pow += 1
end
nums[0...count]
end
 
count = 20
puts "First 20 palindromic gapful numbers ending with:"
(1..9).each { |digit| print "#{digit} : #{digitscount(digit, count)}\n" }
 
count = 100
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
(1..9).each { |digit| print "#{digit} : #{digitscount(digit, count).last(15)}\n" }
 
count = 1000
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
(1..9).each { |digit| print "#{digit} : #{digitscount(digit, count).last(10)}\n" }

Orders of Magnitude Faster: Direct Generation of Numbers[edit]

Crystal is a statically typed and a compiled language.
The code as implemented has been tested to produce optimum performance.

System: I7-6700HQ, 3.5GHz, Linux Kernel 5.6.17, Crystal 0.35
Run as: $ crystal run --release palindromicgapfuls.cr 

Optimized version using number<->string conversion: 21.5 secs

def palindromicgapfuls(digit, count)
gapfuls = [] of UInt64 # array of palindromic gapfuls
dd = 11_u64 * digit # digit gapful divisor: 11, 22,...88, 99
(2..).select do |power|
base = 10_u64**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_u64
10.times do
gapfuls << palindrome if palindrome.divisible_by?(dd)
return gapfuls if gapfuls.size == count
palindrome += base11
end
else
palindrome = (left_half.rchop + right_half).to_u64
10.times do
gapfuls << palindrome if palindrome.divisible_by?(dd)
return gapfuls if gapfuls.size == count
palindrome += base
end
end
end
end
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
puts (Time.monotonic - start).total_seconds

Compact version: 22.0 secs

def palindromicgapfuls(digit, count)
gapfuls = [] of UInt64 # array of palindromic gapfuls
dd = 11_u64 * digit # digit gapful divisor: 11, 22,...88, 99
(2..).select do |power|
base = 10_u64**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
palindrome, left_half = 0_u64, front_half.to_s
basep, right_half = base11, left_half.reverse
(basep = base; left_half = left_half.rchop) if power.even?
palindrome = (left_half + right_half).to_u64
10.times do
gapfuls << palindrome if palindrome.divisible_by?(dd)
return gapfuls if gapfuls.size == count
palindrome += basep
end
end
end
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
puts (Time.monotonic - start).total_seconds

Object Oriented implementation: same speed - 21.8 seconds
Here using a Struct (allocated on stack) is faster than using a Class (allocated on heap)

struct PalindromicGapfuls
include Enumerable(UInt64)
 
@dd : Int32
 
def initialize(@digit : Int32)
@dd = 11 * @digit # digit gapful divisor: 11, 22,...88, 99
end
 
def each
(2..).select do |power|
base = 10_u64**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * @digit # starting half for this digit: 10.. to 90..
next_lo = base * (@digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_u64
10.times do
yield palindrome if palindrome.divisible_by?(@dd)
palindrome += base11
end
else
palindrome = (left_half.rchop + right_half).to_u64
10.times do
yield palindrome if palindrome.divisible_by?(@dd)
palindrome += base
end
end
end
end
end
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
puts (Time.monotonic - start).total_seconds

Original version optimized for minimal memory use: 24.6 secs

def palindromicgapfuls(digit, count, keep)
palcnt = 0 # count of gapful palindromes
to_skip = count - keep # count of unwanted values to skip
gapfuls = [] of UInt64 # array of palindromic gapfuls
dd = 11_u64 * digit # digit gapful divisor: 11, 22,...88, 99
(2..).select do |power|
base = 10_u64**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_u64
10.times do
gapfuls << palindrome if palindrome.divisible_by?(dd) && (palcnt += 1) > to_skip
palindrome += base11
end
else
palindrome = (left_half.rchop + right_half).to_u64
10.times do
gapfuls << palindrome if palindrome.divisible_by?(dd) && (palcnt += 1) > to_skip
palindrome += base
end
end
return gapfuls[0...keep] unless gapfuls.size < keep
end
end
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
puts (Time.monotonic - start).total_seconds

Compact version optimized for minimal memory use: 24.5 secs

def palindromicgapfuls(digit, count, keep)
palcnt = 0 # count of gapful palindromes
to_skip = count - keep # count of unwanted values to skip
gapfuls = [] of UInt64 # array of palindromic gapfuls
dd, base = 11_u64 * digit, 1_u64 # digit gapful divisor: 11, 22,...88, 99
(2..).select do |power|
base *= 10 if power.even? # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
palindrome, left_half = 0_u64, front_half.to_s
basep, right_half = base11, left_half.reverse
(basep = base; left_half = left_half.rchop) if power.even?
palindrome = (left_half + right_half).to_u64
10.times do
gapfuls << palindrome if palindrome.divisible_by?(dd) && (palcnt += 1) > to_skip
palindrome += basep
end
return gapfuls[0...keep] unless palcnt < count
end
end
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
puts (Time.monotonic - start).total_seconds

OOP version optimized for minimal memory use - 25.4 secs
It creates an output method that skips the unwanted values and only keeps/stores the desired ones.

struct PalindromicGapfuls
include Enumerable(UInt64)
 
@dd : Int32
 
def initialize(@digit : Int32)
@dd = 11 * @digit # digit gapful divisor: 11, 22,...88, 99
end
 
def each
(2..).select do |power|
base = 10_u64**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * @digit # starting half for this digit: 10.. to 90..
next_lo = base * (@digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_u64
10.times do
yield palindrome if palindrome.divisible_by?(@dd)
palindrome += base11
end
else
palindrome = (left_half.rchop + right_half).to_u64
10.times do
yield palindrome if palindrome.divisible_by?(@dd)
palindrome += base
end
end
end
end
end
 
# Optimized output method: only keep desired values.
def keep_from(count, keep)
to_skip = (count - keep)
kept = [] of UInt64
each_with_index do |value, i|
i < to_skip ? next : kept << value
return kept if kept.size == keep
end
end
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
puts (Time.monotonic - start).total_seconds

Compact minimized memory version that numerically create palindromes: 9.2 secs

def palindromicgapfuls(digit, count, keep)
palcnt = 0 # count of gapful palindromes
to_skip = count - keep # count of unwanted values to skip
gapfuls = [] of UInt64 # array of palindromic gapfuls
dd, base = 11_u64 * digit, 1_u64 # digit gapful divisor: 11, 22,...88, 99
(2..).select do |power|
base *= 10 if power.even? # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
basep = power.odd? ? base11 : base
palindrome = make_palindrome(front_half, power)
10.times do
gapfuls << palindrome if palindrome.divisible_by?(dd) && (palcnt += 1) > to_skip
palindrome += basep
end
return gapfuls[0...keep] unless palcnt < count
end
end
end
 
def make_palindrome(front_half, power)
result = front_half
result //= 10 if power.even?
while front_half > 0
result = result * 10 + front_half.remainder(10)
front_half //= 10
end
result
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
puts (Time.monotonic - start).total_seconds

OOP version optimized for minimal memory use, palindromes created numerically - 9.59 secs

struct PalindromicGapfuls
include Enumerable(UInt64)
 
@dd : Int32
 
def initialize(@digit : Int32)
@dd = 11 * @digit # digit gapful divisor: 11, 22,...88, 99
end
 
def each
(2..).select do |power|
base = 10_u64**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * @digit # starting half for this digit: 10.. to 90..
next_lo = base * (@digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
basep = power.odd? ? base11 : base
palindrome = make_palindrome(front_half, power)
10.times do
yield palindrome if palindrome.divisible_by?(@dd)
palindrome += basep
end
end
end
end
 
# Optimized output method: only keep desired values.
def keep_from(count, keep)
to_skip = (count - keep)
kept = [] of UInt64
each_with_index do |value, i|
i < to_skip ? next : kept << value
return kept if kept.size == keep
end
end
 
def make_palindrome(front_half, power)
result = front_half
result //= 10 if power.even?
while front_half > 0
result = result * 10 + front_half.remainder(10)
front_half //= 10
end
result
end
end
 
start = Time.monotonic
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
puts (Time.monotonic - start).total_seconds
Output:
First 20 palindromic gapful numbers 100 ending with:
1 : [121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
2 : [242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
3 : [363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
4 : [484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
5 : [5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
6 : [6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
7 : [7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
8 : [8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
9 : [9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]

Last 15 of first 100 palindromic gapful numbers ending in:
1 : [165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
2 : [265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
3 : [30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
4 : [4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
5 : [565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
6 : [60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
7 : [72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
8 : [80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
9 : [95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]

Last 10 of first 1000 palindromic gapful numbers ending in:
1 : [17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
2 : [27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
3 : [3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
4 : [482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
5 : [57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
6 : [6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
7 : [7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
8 : [8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
9 : [9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]

100,000th palindromic gapful number ending with:
1 : [178788887871]
2 : [278788887872]
3 : [30878611687803]
4 : [4833326233384]
5 : [578789987875]
6 : [60878611687806]
7 : [74826144162847]
8 : [80869688696808]
9 : [96878077087869]

1,000,000th palindromic gapful number ending with:
1 : [17878799787871]
2 : [27878799787872]
3 : [3087876666787803]
4 : [483333272333384]
5 : [57878799787875]
6 : [6087876996787806]
7 : [7487226666227847]
8 : [8086969559696808]
9 : [9687870990787869]

10,000,000th palindromic gapful number ending with:
1 : [1787878888787871]
2 : [2787878888787872]
3 : [308787855558787803]
4 : [48333332623333384]
5 : [5787878998787875]
6 : [608787855558787806]
7 : [748867523325768847]
8 : [808696968869696808]
9 : [968787783387787869]

Translation of F#[edit]

Translation of: Ruby of F#


Recursive version; max memory consumption hits ~22%; comment out necessary outputs to run.
May produce: "GC Warning: Repeated allocation of very large block (appr. size xxxxxx):" messages in output.
For Crystal >= 0.34, the operations &+, &*, and &** turnoff default compiler overflow checks.

System: I7-6700HQ, 3.5GHz, 16GB, Linux Kernel 5.9.10, Crystal 0.35.1
Run as: $ crystal run --release fsharp2crystal.cr
Best Time: 29.455717914 secs
class PalNo
@digit : UInt64
@dd : UInt64
 
def initialize(digit : Int32)
@digit, @l, @dd = digit.to_u64, 3, 11u64 * digit
end
 
def fN(n : Int32)
return [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] of UInt64 if n == 1
return [0, 11, 22, 33, 44, 55, 66, 77, 88, 99] of UInt64 if n == 2
a = [] of UInt64
([0, 1, 2, 3, 4, 5, 6, 7, 8, 9] of UInt64).product(fN(n - 2)) do |g0, g1|
a << g0.to_u64 &* 10u64 &** (n - 1) &+ g0.to_u64 &+ 10u64 &* g1.to_u64
end
return a
end
 
def show(count, keep)
to_skip, palcnt, pals = count - keep, 0, [] of UInt64
while palcnt < count
fN(@l - 2).each do |g|
pal = @digit * 10u64 &** (@l - 1) + @digit + 10u64 &* g
pals << pal if pal % @dd == 0 && (palcnt += 1) > to_skip
break if palcnt - to_skip == keep
end
@l += 1
end
print pals; puts
end
end
 
start = Time.monotonic
 
(1..9).each { |digit| PalNo.new(digit).show(20, 20) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(100, 15) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(1000, 10) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(100_000, 1) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(1_000_000, 1) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(10_000_000, 1) }; puts "####"
 
puts (Time.monotonic - start).total_seconds
 
Output:
[121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
[242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
[363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
[484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
[5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
[6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
[7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
[8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
[9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]
####
[165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
[265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
[30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
[4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
[565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
[60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
[72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
[80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
[95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]
####
[17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
[27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
[3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
[482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
[57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
[6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
[7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
[8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
[9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]
####
[178788887871]
[278788887872]
[30878611687803]
[4833326233384]
[578789987875]
[60878611687806]
[74826144162847]
[80869688696808]
[96878077087869]
####
[17878799787871]
[27878799787872]
[3087876666787803]
[483333272333384]
[57878799787875]
[6087876996787806]
[7487226666227847]
[8086969559696808]
[9687870990787869]
####
[1787878888787871]
[2787878888787872]
[308787855558787803]
[48333332623333384]
[5787878998787875]
[608787855558787806]
[748867523325768847]
[808696968869696808]
[968787783387787869]
####

F#[edit]

 
// Palindromic Gapful Numbers . Nigel Galloway: December 3rd., 2020
let rec fN g l=seq{match l with 3->yield! seq{for n in 0L..9L->g*100L+g+n*10L}
|4->yield! seq{for n in 0L..9L->g*1000L+g+n*110L}
|_->yield! seq{for n in 0L..9L do for i in fN n (l-2)->i*10L+g+g*(pown 10L (l-1))}}
 
let rcGf n=let rec rcGf g=seq{yield! fN n g|>Seq.filter(fun g->g%(10L*n+n)=0L); yield! rcGf(g+1)} in rcGf 3
 
[1L..9L]|>Seq.iter(fun n->rcGf n|>Seq.take 20|>Seq.iter(printf "%d ");printfn "");printfn "#####"
[1L..9L]|>Seq.iter(fun n->rcGf n|>Seq.skip 85|>Seq.take 15|>Seq.iter(printf "%d ");printfn "");printfn "#####"
[1L..9L]|>Seq.iter(fun n->rcGf n|>Seq.skip 990|>Seq.take 10|>Seq.iter(printf "%d ");printfn "");printfn "#####"
 
Output:
121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069
#####
165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284 
565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569
#####
17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871
27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872
3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384
57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875
6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869
#####

Factor[edit]

USING: formatting fry io kernel lists lists.lazy locals math
math.functions math.ranges math.text.utils prettyprint sequences ;
IN: rosetta-code.palindromic-gapful-numbers
 
! Palindromic numbers are relatively rare compared to gapful
! numbers, so our strategy for finding palindromic gapful
! numbers is to filter gapful numbers from palindromic numbers.
 
! Palindromic numbers can be generated directly rather than
! filtered or identified from the natural numbers. This is a
! significant speedup since palindromic numbers are relatively
! rare in the natural numbers.
 
! Here I have used a generation method similar to
! https://www.geeksforgeeks.org/generate-palindromic-numbers-less-n/
 
! Create a palindrome from its base natural number.
! e.g. 321 t -> 32123
! 321 f -> 321123
: create-palindrome ( n odd? -- m )
dupd [ 10 /i ] when swap [ over 0 > ]
[ 10 * [ 10 /mod ] [ + ] bi* ] while nip ;
 
! Create an infinite lazy list of palindromic numbers starting
! at 100.
: palindromes ( -- l )
1 lfrom [
10 swap ^ dup 10 * [a,b)
[ [ t create-palindrome ] map ]
[ [ f create-palindrome ] map ] bi
[ sequence>list ] [email protected] lappend
] lmap-lazy lconcat ;
 
! Is an integer gapful?
: gapful? ( n -- ? )
dup 1 digit-groups [ first ] [ last 10 * + ] bi divisor? ;
 
! Create an infinite lazy list of gapful palindromes.
: gapful-palindromes ( -- l ) palindromes [ gapful? ] lfilter ;
 
:: show-palindromic-gapfuls ( last of -- )
gapful-palindromes :> nums
last of
"~~==[ Last  %d of  %d palindromic gapful numbers starting at 100 ]==~~\n"
printf 9 [1,b] [| d |
of nums [ 10 mod d = ] lfilter ltake list>array
last tail* d pprint ": " write [ pprint bl ] each nl
] each nl ;
 
20 20  ! part 1
15 100  ! part 2
10 1000  ! part 3 (Optional)
[ show-palindromic-gapfuls ] [email protected]
Output:
~~==[ Last  20  of  20  palindromic gapful numbers starting at 100 ]==~~
1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591 
2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392 
3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333 
4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804 
5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895 
6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336 
7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087 
8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648 
9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069 

~~==[ Last  15  of  100  palindromic gapful numbers starting at 100 ]==~~
1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971 
2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972 
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703 
4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284 
5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975 
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806 
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927 
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808 
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569 

~~==[ Last  10  of  1000  palindromic gapful numbers starting at 100 ]==~~
1: 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871 
2: 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872 
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803 
4: 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384 
5: 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875 
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806 
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547 
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808 
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869 

FreeBASIC[edit]

Time-consuming brute-force solution with only a few speedups...

function is_gapful( n as uinteger ) as boolean
if n<100 then return false
dim as string ns = str(n)
dim as uinteger gap = 10*val(mid(ns,1,1)) + val(mid(ns,len(ns),1))
if n mod gap = 0 then return true else return false
end function
 
function is_palindrome( n as uinteger ) as boolean
dim as string ns = str(n)
for i as uinteger = 1 to len(ns)\2
if mid(ns,i,1) <> mid(ns,len(ns)+1-i,1) then return false
next i
return true
end function
 
function padto( n as uinteger, s as integer ) as string
dim as string outstr=""
dim as integer k = len(str(n))
for i as integer = 1 to s-k
outstr = " " + outstr
next i
return outstr + str(n)
end function
 
sub print_range( yays() as uinteger, first as uinteger, last as uinteger)
dim as string outstr
for i as uinteger = first to last
outstr = padto(i,4)+"  : "
for d as uinteger = 1 to 9
outstr += padto(yays(d,i), 11)
next d
print outstr
next i
end sub
 
#define is_yay(n) (is_gapful(n) and is_palindrome(n))
#define log10(n) log(n)*0.43429448190325182765112891891660508229
 
dim as uinteger yays(1 to 9, 1 to 1000), nyays(1 to 9), num = 99, fd
do
num += 1 : fd = val(left(str(num),1))
if fd = 0 then continue do 'no paligap will have 0 as leading digit
if nyays(fd) = 1000 then
num = (fd+1)*10^int(log10(num))
end if
if is_yay(num) then
nyays(fd) += 1
yays(fd, nyays(fd)) = num
end if
for y as uinteger = 1 to 9
if nyays(y) < 1000 then continue do
next y
exit do
loop
 
'excessive output requirements for such a simple task
print_range(yays(), 1, 20)
print_range(yays(), 86, 100)
print_range(yays(), 991, 1000)
Output:
   1  :          121        242        363        484       5005       6006       7007       8008       9009
   2  :         1001       2002       3003       4004       5115       6336       7777       8448       9999
   3  :         1111       2112       3333       4224       5225       6666      77077       8888      94149
   4  :         1221       2222       3663       4444       5335       6996     700007      80608      99099
   5  :         1331       2332       3993       4664       5445      61116     707707      86768     900009
   6  :         1441       2442      31713       4884       5555      64746     710017      88088     909909
   7  :         1551       2552      33033      40304       5665      66066     717717     800008     918819
   8  :         1661       2662      36663      42724       5775      69696     720027     802208     927729
   9  :         1771       2772     300003      44044       5885     600006     727727     804408     936639
  10  :         1881       2882     303303      46464       5995     603306     730037     806608     945549
  11  :         1991       2992     306603      48884      50105     606606     737737     808808     954459
  12  :        10901      20702     309903     400004      51315     609906     740047     821128     963369
  13  :        11011      21912     312213     401104      52525     612216     747747     823328     972279
  14  :        12221      22022     315513     402204      53735     615516     750057     825528     981189
  15  :        13431      23232     318813     403304      54945     618816     757757     827728     990099
  16  :        14641      24442     321123     404404      55055     621126     760067     829928     999999
  17  :        15851      25652     324423     405504      56265     624426     767767     840048    9459549
  18  :        17171      26862     327723     406604      57475     627726     770077     842248    9508059
  19  :        18381      28182     330033     407704      58685     630036     777777     844448    9557559
  20  :        19591      29392     333333     408804      59895     633336     780087     846648    9606069
  86  :       165561     265562   30366303    4473744     565565   60399306   72299227   80611608   95311359
  87  :       166661     266662   30399303    4485844     566665   60422406   72322327   80622608   95400459
  88  :       167761     267762   30422403    4497944     567765   60455406   72399327   80633608   95499459
  89  :       168861     268862   30455403    4607064     568865   60488406   72422427   80644608   95588559
  90  :       169961     269962   30488403    4619164     569965   60511506   72499427   80655608   95677659
  91  :       170071     270072   30511503    4620264     570075   60544506   72522527   80666608   95766759
  92  :       171171     271172   30544503    4632364     571175   60577506   72599527   80677608   95855859
  93  :       172271     272272   30577503    4644464     572275   60600606   72622627   80688608   95944959
  94  :       173371     273372   30600603    4656564     573375   60633606   72699627   80699608   96033069
  95  :       174471     274472   30633603    4668664     574475   60666606   72722727   80800808   96122169
  96  :       175571     275572   30666603    4681864     575575   60699606   72799727   80811808   96211269
  97  :       176671     276672   30699603    4693964     576675   60722706   72822827   80822808   96300369
  98  :       177771     277772   30722703    4803084     577775   60755706   72899827   80833808   96399369
  99  :       178871     278872   30755703    4815184     578875   60788706   72922927   80844808   96488469
 100  :       179971     279972   30788703    4827284     579975   60811806   72999927   80855808   96577569
 991  :     17799771   27799772 3084004803  482282284   57800875 6084004806 7452992547 8085995808 9675005769
 992  :     17800871   27800872 3084334803  482414284   57811875 6084334806 7453223547 8086006808 9675995769
 993  :     17811871   27811872 3084664803  482535284   57822875 6084664806 7453993547 8086116808 9676886769
 994  :     17822871   27822872 3084994803  482656284   57833875 6084994806 7454224547 8086226808 9677777769
 995  :     17833871   27833872 3085225803  482777284   57844875 6085225806 7454994547 8086336808 9678668769
 996  :     17844871   27844872 3085555803  482898284   57855875 6085555806 7455225547 8086446808 9679559769
 997  :     17855871   27855872 3085885803  482909284   57866875 6085885806 7455995547 8086556808 9680440869
 998  :     17866871   27866872 3086116803  483020384   57877875 6086116806 7456226547 8086666808 9681331869
 999  :     17877871   27877872 3086446803  483141384   57888875 6086446806 7456996547 8086776808 9682222869
1000  :     17888871   27888872 3086776803  483262384   57899875 6086776806 7457227547 8086886808 9683113869

Fōrmulæ[edit]

In this page you can see the solution of this task.

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.

The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.

Go[edit]

This uses the same strategy as the Factor entry i.e. to generate all palindromic numbers in order and then test whether they're gapful or not.

To keep the Pascal entry company, I've extended the search to the first 10 million such numbers for each of the nine sets.

package main
 
import "fmt"
 
func reverse(s uint64) uint64 {
e := uint64(0)
for s > 0 {
e = e*10 + (s % 10)
s /= 10
}
return e
}
 
func commatize(n uint) string {
s := fmt.Sprintf("%d", n)
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}
 
func ord(n uint) string {
var suffix string
if n > 10 && ((n-11)%100 == 0 || (n-12)%100 == 0 || (n-13)%100 == 0) {
suffix = "th"
} else {
switch n % 10 {
case 1:
suffix = "st"
case 2:
suffix = "nd"
case 3:
suffix = "rd"
default:
suffix = "th"
}
}
return fmt.Sprintf("%s%s", commatize(n), suffix)
}
 
func main() {
const max = 10_000_000
data := [][3]uint{{1, 20, 7}, {86, 100, 8}, {991, 1000, 10}, {9995, 10000, 12}, {1e5, 1e5, 14},
{1e6, 1e6, 16}, {1e7, 1e7, 18}}
results := make(map[uint][]uint64)
for _, d := range data {
for i := d[0]; i <= d[1]; i++ {
results[i] = make([]uint64, 9)
}
}
var p uint64
outer:
for d := uint64(1); d < 10; d++ {
count := uint(0)
pow := uint64(1)
fl := d * 11
for nd := 3; nd < 20; nd++ {
slim := (d + 1) * pow
for s := d * pow; s < slim; s++ {
e := reverse(s)
mlim := uint64(1)
if nd%2 == 1 {
mlim = 10
}
for m := uint64(0); m < mlim; m++ {
if nd%2 == 0 {
p = s*pow*10 + e
} else {
p = s*pow*100 + m*pow*10 + e
}
if p%fl == 0 {
count++
if _, ok := results[count]; ok {
results[count][d-1] = p
}
if count == max {
continue outer
}
}
}
}
if nd%2 == 1 {
pow *= 10
}
}
}
 
for _, d := range data {
if d[0] != d[1] {
fmt.Printf("%s to %s palindromic gapful numbers (> 100) ending with:\n", ord(d[0]), ord(d[1]))
} else {
fmt.Printf("%s palindromic gapful number (> 100) ending with:\n", ord(d[0]))
}
for i := 1; i <= 9; i++ {
fmt.Printf("%d: ", i)
for j := d[0]; j <= d[1]; j++ {
fmt.Printf("%*d ", d[2], results[j][i-1])
}
fmt.Println()
}
fmt.Println()
}
}
Output:
1st to 20th palindromic gapful numbers (> 100) ending with:
1:     121    1001    1111    1221    1331    1441    1551    1661    1771    1881    1991   10901   11011   12221   13431   14641   15851   17171   18381   19591 
2:     242    2002    2112    2222    2332    2442    2552    2662    2772    2882    2992   20702   21912   22022   23232   24442   25652   26862   28182   29392 
3:     363    3003    3333    3663    3993   31713   33033   36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333 
4:     484    4004    4224    4444    4664    4884   40304   42724   44044   46464   48884  400004  401104  402204  403304  404404  405504  406604  407704  408804 
5:    5005    5115    5225    5335    5445    5555    5665    5775    5885    5995   50105   51315   52525   53735   54945   55055   56265   57475   58685   59895 
6:    6006    6336    6666    6996   61116   64746   66066   69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336 
7:    7007    7777   77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087 
8:    8008    8448    8888   80608   86768   88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648 
9:    9009    9999   94149   99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999 9459549 9508059 9557559 9606069 

86th to 100th palindromic gapful numbers (> 100) ending with:
1:   165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971 
2:   265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972 
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703 
4:  4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284 
5:   565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975 
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806 
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927 
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808 
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569 

991st to 1,000th palindromic gapful numbers (> 100) ending with:
1:   17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871 
2:   27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872 
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803 
4:  482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384 
5:   57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875 
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806 
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547 
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808 
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869 

9,995th to 10,000th palindromic gapful numbers (> 100) ending with:
1:   1787447871   1787557871   1787667871   1787777871   1787887871   1787997871 
2:   2787447872   2787557872   2787667872   2787777872   2787887872   2787997872 
3: 308757757803 308760067803 308763367803 308766667803 308769967803 308772277803 
4:  48326662384  48327872384  48329192384  48330303384  48331513384  48332723384 
5:   5787447875   5787557875   5787667875   5787777875   5787887875   5787997875 
6: 608760067806 608763367806 608766667806 608769967806 608772277806 608775577806 
7: 746951159647 746958859647 746961169647 746968869647 746971179647 746978879647 
8: 808690096808 808691196808 808692296808 808693396808 808694496808 808695596808 
9: 968688886869 968697796869 968706607869 968715517869 968724427869 968733337869 

100,000th palindromic gapful number (> 100) ending with:
1:   178788887871 
2:   278788887872 
3: 30878611687803 
4:  4833326233384 
5:   578789987875 
6: 60878611687806 
7: 74826144162847 
8: 80869688696808 
9: 96878077087869 

1,000,000th palindromic gapful number (> 100) ending with:
1:   17878799787871 
2:   27878799787872 
3: 3087876666787803 
4:  483333272333384 
5:   57878799787875 
6: 6087876996787806 
7: 7487226666227847 
8: 8086969559696808 
9: 9687870990787869 

10,000,000th palindromic gapful number (> 100) ending with:
1:   1787878888787871 
2:   2787878888787872 
3: 308787855558787803 
4:  48333332623333384 
5:   5787878998787875 
6: 608787855558787806 
7: 748867523325768847 
8: 808696968869696808 
9: 968787783387787869 

Haskell[edit]

Brute Force[edit]

import Control.Monad (guard)
 
palindromic :: Int -> Bool
palindromic n = d == reverse d
where
d = show n
 
gapful :: Int -> Bool
gapful n = n `rem` firstLastDigit == 0
where
firstLastDigit = read [head asDigits, last asDigits]
asDigits = show n
 
result :: Int -> [Int]
result d = do
x <- [(d+100),(d+110)..]
guard $ palindromic x && gapful x
pure x
 
showSets :: (Int -> String) -> IO ()
showSets r = go 1
where
go n = if n <= 9 then do
putStrLn (show n ++ ": " ++ r n)
go (succ n)
else pure ()
 
main :: IO ()
main = do
putStrLn "\nFirst 20 palindromic gapful numbers ending in:"
showSets (show . take 20 . result)
putStrLn "\nLast 15 of first 100 palindromic gapful numbers ending in:"
showSets (show . drop 85 . take 100 . result)
putStrLn "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
showSets (show . drop 990 . take 1000 . result)
putStrLn "\ndone."

Optimized[edit]

Here the approach is to generate a series of palindromes.

import Data.List     (sort, unfoldr)
import Control.Monad (guard)
 
gapful :: Int -> Bool
gapful n = n `rem` firstLastDigit n == 0
where
firstLastDigit = (\xs -> head xs * 10 + last xs) . reverse
. unfoldr (\n -> guard (n /= 0) >> pure (n `mod` 10, n `div` 10))
 
toPalinDrome :: Int -> [Int]
toPalinDrome n = filter ((&&) . (> 100) <*> gapful) [go n n, go n (n `div` 10)]
where
go p 0 = p
go p n'' = go (p * 10 + (n'' `mod` 10)) (n'' `div` 10)
 
gapfulPalindromes :: [Int]
gapfulPalindromes = (sort . (=<<) toPalinDrome) [1 .. 99999]
 
endsWith :: Int -> [Int]
endsWith n = filter ((n ==) . (`mod` 10)) gapfulPalindromes
 
showSets :: (String, [Int] -> [Int]) -> String
showSets (k, r) =
k ++
" palindromic gapful numbers ending in:\n" ++
unlines ((<*>) ((++) . show) ((": " ++) . show . r . endsWith) <$> [1 .. 9])
 
main :: IO ()
main =
mapM_
(putStrLn . showSets)
[ ("First 20", take 20)
, ("Last 15 of first 100", drop 85 . take 100)
, ("Last 10 of first 1000", drop 990 . take 1000)
]
Output:
First 20 palindromic gapful numbers ending in:
1: [121,1001,1111,1221,1331,1441,1551,1661,1771,1881,1991,10901,11011,12221,13431,14641,15851,17171,18381,19591]
2: [242,2002,2112,2222,2332,2442,2552,2662,2772,2882,2992,20702,21912,22022,23232,24442,25652,26862,28182,29392]
3: [363,3003,3333,3663,3993,31713,33033,36663,300003,303303,306603,309903,312213,315513,318813,321123,324423,327723,330033,333333]
4: [484,4004,4224,4444,4664,4884,40304,42724,44044,46464,48884,400004,401104,402204,403304,404404,405504,406604,407704,408804]
5: [5005,5115,5225,5335,5445,5555,5665,5775,5885,5995,50105,51315,52525,53735,54945,55055,56265,57475,58685,59895]
6: [6006,6336,6666,6996,61116,64746,66066,69696,600006,603306,606606,609906,612216,615516,618816,621126,624426,627726,630036,633336]
7: [7007,7777,77077,700007,707707,710017,717717,720027,727727,730037,737737,740047,747747,750057,757757,760067,767767,770077,777777,780087]
8: [8008,8448,8888,80608,86768,88088,800008,802208,804408,806608,808808,821128,823328,825528,827728,829928,840048,842248,844448,846648]
9: [9009,9999,94149,99099,900009,909909,918819,927729,936639,945549,954459,963369,972279,981189,990099,999999,9459549,9508059,9557559,9606069]

Last 15 of first 100 palindromic gapful numbers ending in:
1: [165561,166661,167761,168861,169961,170071,171171,172271,173371,174471,175571,176671,177771,178871,179971]
2: [265562,266662,267762,268862,269962,270072,271172,272272,273372,274472,275572,276672,277772,278872,279972]
3: [30366303,30399303,30422403,30455403,30488403,30511503,30544503,30577503,30600603,30633603,30666603,30699603,30722703,30755703,30788703]
4: [4473744,4485844,4497944,4607064,4619164,4620264,4632364,4644464,4656564,4668664,4681864,4693964,4803084,4815184,4827284]
5: [565565,566665,567765,568865,569965,570075,571175,572275,573375,574475,575575,576675,577775,578875,579975]
6: [60399306,60422406,60455406,60488406,60511506,60544506,60577506,60600606,60633606,60666606,60699606,60722706,60755706,60788706,60811806]
7: [72299227,72322327,72399327,72422427,72499427,72522527,72599527,72622627,72699627,72722727,72799727,72822827,72899827,72922927,72999927]
8: [80611608,80622608,80633608,80644608,80655608,80666608,80677608,80688608,80699608,80800808,80811808,80822808,80833808,80844808,80855808]
9: [95311359,95400459,95499459,95588559,95677659,95766759,95855859,95944959,96033069,96122169,96211269,96300369,96399369,96488469,96577569]

Last 10 of first 1000 palindromic gapful numbers ending in:
1: [17799771,17800871,17811871,17822871,17833871,17844871,17855871,17866871,17877871,17888871]
2: [27799772,27800872,27811872,27822872,27833872,27844872,27855872,27866872,27877872,27888872]
3: [3084004803,3084334803,3084664803,3084994803,3085225803,3085555803,3085885803,3086116803,3086446803,3086776803]
4: [482282284,482414284,482535284,482656284,482777284,482898284,482909284,483020384,483141384,483262384]
5: [57800875,57811875,57822875,57833875,57844875,57855875,57866875,57877875,57888875,57899875]
6: [6084004806,6084334806,6084664806,6084994806,6085225806,6085555806,6085885806,6086116806,6086446806,6086776806]
7: [7452992547,7453223547,7453993547,7454224547,7454994547,7455225547,7455995547,7456226547,7456996547,7457227547]
8: [8085995808,8086006808,8086116808,8086226808,8086336808,8086446808,8086556808,8086666808,8086776808,8086886808]
9: [9675005769,9675995769,9676886769,9677777769,9678668769,9679559769,9680440869,9681331869,9682222869,9683113869]

done.

J[edit]

Part 1:

   task1 =: {. (((= 10&#:) # ]) palindromic_multiples_of_eleven)
   palindromic_multiples_of_eleven =: [: (#~ (99&< *. palindrome&>)) (11*i.100001)&*
   palindrome =: (-: |.)@:":

   20 task1&> >:i.9
 121 1001  1111   1221   1331   1441   1551   1661   1771   1881   1991  10901  11011  12221  13431  14641   15851   17171   18381   19591
 242 2002  2112   2222   2332   2442   2552   2662   2772   2882   2992  20702  21912  22022  23232  24442   25652   26862   28182   29392
 363 3003  3333   3663   3993  31713  33033  36663 300003 303303 306603 309903 312213 315513 318813 321123  324423  327723  330033  333333
 484 4004  4224   4444   4664   4884  40304  42724  44044  46464  48884 400004 401104 402204 403304 404404  405504  406604  407704  408804
5005 5115  5225   5335   5445   5555   5665   5775   5885   5995  50105  51315  52525  53735  54945  55055   56265   57475   58685   59895
6006 6336  6666   6996  61116  64746  66066  69696 600006 603306 606606 609906 612216 615516 618816 621126  624426  627726  630036  633336
7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067  767767  770077  777777  780087
8008 8448  8888  80608  86768  88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928  840048  842248  844448  846648
9009 9999 94149  99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

Part 2:

 
palindromify=: [: , ((,~ |[email protected]}.) ; (,~ |.))&>
gapful=: (0 = (|~ ({.,{:)&.(10&#.inv)))&>
task2_cartesian_products=: [: , [: { ((i. 10) ; (>: i. 9)) #~ ,&1
task2_palindromes=: [: 10&#.&> [: palindromify task2_cartesian_products
task2_gapfuls=: [: /:~ [: ; [: (#~ gapful)@task2_palindromes&.> >:@i.
 
   palindromify { ;: 'abc XY'   NB. demonstration
+---+----+---+----+---+----+---+----+---+----+---+----+
|XaX|XaaX|YaY|YaaY|XbX|XbbX|YbY|YbbY|XcX|XccX|YcY|YccY|
+---+----+---+----+---+----+---+----+---+----+---+----+


   NB. task2 solution

   A=: task2_gapfuls 4  NB. A is an ordered vector of the 3 to 10 digit gapful palindromes
   B=: (</.~ 10&#:) A   NB. B are A grouped by last (first) digit

   (# , {:)&> B  NB. tally and tail of each group
12120 1999999991
12120 2999999992
 4044 3999999993
 6061 4899999984
12120 5999999995
 4044 6999999996
 1785 7999449997
 3031 8889999888
 1352 9999999999

   (_15 {. 100&{.)&> B  NB. the last 15 of the first hundred
  165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971
  265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972
30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
 4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284
  565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975
60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

   (_10 {. 1000&{.)&> B  NB. the last 10 of the first 1000
  17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871
  27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872
3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
 482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384
  57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875
6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869


   NB. timing
   NB. B matches the rearranged expression
   timespacex'assert B -: (</.~ 10&#:) task2_gapfuls 4'  NB. approximate timing for the substantial part of the effort
0.551638 7.2343e7

   NB. full memory, Thinkpad W540

   JVERSION
Engine: j901/j64avx2/windows
Release-c: commercial/2020-01-11T13:29:14
Library: 9.01.20
Platform: Win 64
Installer: J901 install
InstallPath: c:/program files/j901
Contact: www.jsoftware.com

Java[edit]

 
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
 
public class PalindromicGapfulNumbers {
 
public static void main(String[] args) {
System.out.println("First 20 palindromic gapful numbers ending in:");
displayMap(getPalindromicGapfulEnding(20, 20));
 
System.out.printf("%nLast 15 of first 100 palindromic gapful numbers ending in:%n");
displayMap(getPalindromicGapfulEnding(15, 100));
 
System.out.printf("%nLast 10 of first 1000 palindromic gapful numbers ending in:%n");
displayMap(getPalindromicGapfulEnding(10, 1000));
}
 
private static void displayMap(Map<Integer,List<Long>> map) {
for ( int key = 1 ; key <= 9 ; key++ ) {
System.out.println(key + " : " + map.get(key));
}
}
 
public static Map<Integer,List<Long>> getPalindromicGapfulEnding(int countReturned, int firstHowMany) {
Map<Integer,List<Long>> map = new HashMap<>();
Map<Integer,Integer> mapCount = new HashMap<>();
for ( int i = 1 ; i <= 9 ; i++ ) {
map.put(i, new ArrayList<>());
mapCount.put(i, 0);
}
boolean notPopulated = true;
for ( long n = 101 ; notPopulated ; n = nextPalindrome(n) ) {
if ( isGapful(n) ) {
int index = (int) (n % 10);
if ( mapCount.get(index) < firstHowMany ) {
map.get(index).add(n);
mapCount.put(index, mapCount.get(index) + 1);
if ( map.get(index).size() > countReturned ) {
map.get(index).remove(0);
}
}
boolean finished = true;
for ( int i = 1 ; i <= 9 ; i++ ) {
if ( mapCount.get(i) < firstHowMany ) {
finished = false;
break;
}
}
if ( finished ) {
notPopulated = false;
}
}
}
return map;
}
 
public static boolean isGapful(long n) {
String s = Long.toString(n);
return n % Long.parseLong("" + s.charAt(0) + s.charAt(s.length()-1)) == 0;
}
 
public static int length(long n) {
int length = 0;
while ( n > 0 ) {
length += 1;
n /= 10;
}
return length;
}
 
public static long nextPalindrome(long n) {
int length = length(n);
if ( length % 2 == 0 ) {
length /= 2;
while ( length > 0 ) {
n /= 10;
length--;
}
n += 1;
if ( powerTen(n) ) {
return Long.parseLong(n + reverse(n/10));
}
return Long.parseLong(n + reverse(n));
}
length = (length - 1) / 2;
while ( length > 0 ) {
n /= 10;
length--;
}
n += 1;
if ( powerTen(n) ) {
return Long.parseLong(n + reverse(n/100));
}
return Long.parseLong(n + reverse(n/10));
}
 
private static boolean powerTen(long n) {
while ( n > 9 && n % 10 == 0 ) {
n /= 10;
}
return n == 1;
}
 
private static String reverse(long n) {
return (new StringBuilder(n + "")).reverse().toString();
}
 
}
 
Output:
First 20 palindromic gapful numbers ending in:
1 : [121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
2 : [242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
3 : [363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
4 : [484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
5 : [5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
6 : [6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
7 : [7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
8 : [8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
9 : [9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]

Last 15 of first 100 palindromic gapful numbers ending in:
1 : [165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
2 : [265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
3 : [30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
4 : [4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
5 : [565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
6 : [60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
7 : [72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
8 : [80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
9 : [95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]

Last 10 of first 1000 palindromic gapful numbers ending in:
1 : [17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
2 : [27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
3 : [3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
4 : [482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
5 : [57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
6 : [6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
7 : [7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
8 : [8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
9 : [9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]

Julia[edit]

import Base.iterate, Base.IteratorSize, Base.IteratorEltype
 
struct Palindrome x1::UInt8; x2::UInt8; outer::UInt8; end
Base.IteratorSize(p::Palindrome) = Base.IsInfinite()
Base.IteratorEltype(g::Palindrome) = Vector{Int8}
 
function Base.iterate(p::Palindrome, state=(UInt8[p.x1]))
arr, len = [p.outer; state; p.outer], length(state)
if all(c -> c == p.x2, state)
return arr, fill(p.x1, len + 1)
end
for i in (len+1)÷2:-1:1
if state[i] < p.x2
state[len - i + 1] = state[i] = state[i] + one(UInt8)
return arr, state
else
state[len - i + 1] = state[i] = p.x1
end
end
state[1] += one(UInt8)
push!(state, state[1])
return arr, state
end
 
asint(s) = foldl((i, j) -> 10i + j, s)
isgapful(a) = mod(asint(a), a[1] * 11) == 0
GapfulPalindrome(i) = Iterators.filter(isgapful, Iterators.take(Palindrome(0, 9, i), 100000000000))
 
function testpal()
for (lastones, outof) in [(20, 20), (15, 100), (10, 1000), (10, 10000), (10, 100000), (10, 1000000), (10, 10000000)]
@time begin
println("\nLast digit | Last $lastones of $outof palindromic gapful numbers from 100\n",
"-----------|----------------------------------------------------------------------------------------------------------------")
output = fill("", 9)
[email protected] for i in 1:9
gplist = sort!(asint.(collect(Iterators.take(GapfulPalindrome(i), outof))))
output[i] = " $i " * string(gplist[end-lastones+1:end]) * "\n"
end
foreach(print, output)
end
end
end
 
testpal()
 
Output:
Last digit | Last 20 of 20 palindromic gapful numbers from 100
-----------|------------------------------------------------------------------------------------------------------------------------
     1        [121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
     2        [242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
     3        [363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
     4        [484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
     5        [5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
     6        [6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
     7        [7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
     8        [8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
     9        [9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]
  0.623229 seconds (1.68 M allocations: 85.926 MiB, 2.02% gc time)

Last digit | Last 15 of 100 palindromic gapful numbers from 100
-----------|------------------------------------------------------------------------------------------------------------------------
     1        [165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
     2        [265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
     3        [30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
     4        [4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
     5        [565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
     6        [60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
     7        [72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
     8        [80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
     9        [95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]
  0.011659 seconds (125.64 k allocations: 4.389 MiB)

Last digit | Last 10 of 1000 palindromic gapful numbers from 100
-----------|------------------------------------------------------------------------------------------------------------------------
     1        [17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
     2        [27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
     3        [3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
     4        [482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
     5        [57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
     6        [6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
     7        [7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
     8        [8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
     9        [9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]
  0.121723 seconds (1.31 M allocations: 45.342 MiB, 23.28% gc time)

Last digit | Last 10 of 10000 palindromic gapful numbers from 100
-----------|------------------------------------------------------------------------------------------------------------------------
     1        [1787007871, 1787117871, 1787227871, 1787337871, 1787447871, 1787557871, 1787667871, 1787777871, 1787887871, 1787997871]
     2        [2787007872, 2787117872, 2787227872, 2787337872, 2787447872, 2787557872, 2787667872, 2787777872, 2787887872, 2787997872]
     3        [308745547803, 308748847803, 308751157803, 308754457803, 308757757803, 308760067803, 308763367803, 308766667803, 308769967803, 308772277803]
     4        [48322922384, 48323032384, 48324242384, 48325452384, 48326662384, 48327872384, 48329192384, 48330303384, 48331513384, 48332723384]
     5        [5787007875, 5787117875, 5787227875, 5787337875, 5787447875, 5787557875, 5787667875, 5787777875, 5787887875, 5787997875]
     6        [608748847806, 608751157806, 608754457806, 608757757806, 608760067806, 608763367806, 608766667806, 608769967806, 608772277806, 608775577806]
     7        [746931139647, 746938839647, 746941149647, 746948849647, 746951159647, 746958859647, 746961169647, 746968869647, 746971179647, 746978879647]
     8        [808686686808, 808687786808, 808688886808, 808689986808, 808690096808, 808691196808, 808692296808, 808693396808, 808694496808, 808695596808]
     9        [968652256869, 968661166869, 968670076869, 968679976869, 968688886869, 968697796869, 968706607869, 968715517869, 968724427869, 968733337869]
  1.194631 seconds (13.03 M allocations: 452.216 MiB, 19.09% gc time)

Last digit | Last 10 of 100000 palindromic gapful numbers from 100
-----------|------------------------------------------------------------------------------------------------------------------------
     1        [178779977871, 178780087871, 178781187871, 178782287871, 178783387871, 178784487871, 178785587871, 178786687871, 178787787871, 178788887871]
     2        [278779977872, 278780087872, 278781187872, 278782287872, 278783387872, 278784487872, 278785587872, 278786687872, 278787787872, 278788887872]
     3        [30878344387803, 30878377387803, 30878400487803, 30878433487803, 30878466487803, 30878499487803, 30878522587803, 30878555587803, 30878588587803, 30878611687803]
     4        [4833228223384, 4833241423384, 4833253523384, 4833265623384, 4833277723384, 4833289823384, 4833290923384, 4833302033384, 4833314133384, 4833326233384]
     5        [578780087875, 578781187875, 578782287875, 578783387875, 578784487875, 578785587875, 578786687875, 578787787875, 578788887875, 578789987875]
     6        [60878344387806, 60878377387806, 60878400487806, 60878433487806, 60878466487806, 60878499487806, 60878522587806, 60878555587806, 60878588587806, 60878611687806]
     7        [74825233252847, 74825333352847, 74825433452847, 74825533552847, 74825633652847, 74825733752847, 74825833852847, 74825933952847, 74826044062847, 74826144162847]
     8        [80869599596808, 80869600696808, 80869611696808, 80869622696808, 80869633696808, 80869644696808, 80869655696808, 80869666696808, 80869677696808, 80869688696808]
     9        [96877266277869, 96877355377869, 96877444477869, 96877533577869, 96877622677869, 96877711777869, 96877800877869, 96877899877869, 96877988977869, 96878077087869]
 10.614688 seconds (129.23 M allocations: 4.349 GiB, 14.81% gc time)

Last digit | Last 10 of 1000000 palindromic gapful numbers from 100
-----------|------------------------------------------------------------------------------------------------------------------------
     1        [17878700787871, 17878711787871, 17878722787871, 17878733787871, 17878744787871, 17878755787871, 17878766787871, 17878777787871, 17878788787871, 17878799787871]
     2        [27878700787872, 27878711787872, 27878722787872, 27878733787872, 27878744787872, 27878755787872, 27878766787872, 27878777787872, 27878788787872, 27878799787872]
     3        [3087873993787803, 3087874224787803, 3087874554787803, 3087874884787803, 3087875115787803, 3087875445787803, 3087875775787803, 3087876006787803, 3087876336787803, 3087876666787803]
     4        [483332292233384, 483332303233384, 483332424233384, 483332545233384, 483332666233384, 483332787233384, 483332919233384, 483333030333384, 483333151333384, 483333272333384]
     5        [57878700787875, 57878711787875, 57878722787875, 57878733787875, 57878744787875, 57878755787875, 57878766787875, 57878777787875, 57878788787875, 57878799787875]
     6        [6087874224787806, 6087874554787806, 6087874884787806, 6087875115787806, 6087875445787806, 6087875775787806, 6087876006787806, 6087876336787806, 6087876666787806, 6087876996787806]
     7        [7487217557127847, 7487218558127847, 7487219559127847, 7487220660227847, 7487221661227847, 7487222662227847, 7487223663227847, 7487224664227847, 7487225665227847, 7487226666227847]
     8        [8086968668696808, 8086968778696808, 8086968888696808, 8086968998696808, 8086969009696808, 8086969119696808, 8086969229696808, 8086969339696808, 8086969449696808, 8086969559696808]
     9        [9687862882687869, 9687863773687869, 9687864664687869, 9687865555687869, 9687866446687869, 9687867337687869, 9687868228687869, 9687869119687869, 9687870000787869, 9687870990787869]
114.847779 seconds (1.28 G allocations: 43.170 GiB, 19.29% gc time)

Last digit | Last 10 of 10000000 palindromic gapful numbers from 100
-----------|------------------------------------------------------------------------------------------------------------------------
     1        [1787877997787871, 1787878008787871, 1787878118787871, 1787878228787871, 1787878338787871, 1787878448787871, 1787878558787871, 1787878668787871, 1787878778787871, 1787878888787871]
     2        [2787877997787872, 2787878008787872, 2787878118787872, 2787878228787872, 2787878338787872, 2787878448787872, 2787878558787872, 2787878668787872, 2787878778787872, 2787878888787872]
     3        [308787828828787803, 308787831138787803, 308787834438787803, 308787837738787803, 308787840048787803, 308787843348787803, 308787846648787803, 308787849948787803, 308787852258787803, 308787855558787803]
     4        [48333322822333384, 48333324142333384, 48333325352333384, 48333326562333384, 48333327772333384, 48333328982333384, 48333329092333384, 48333330203333384, 48333331413333384, 48333332623333384]
     5        [5787878008787875, 5787878118787875, 5787878228787875, 5787878338787875, 5787878448787875, 5787878558787875, 5787878668787875, 5787878778787875, 5787878888787875, 5787878998787875]
     6        [608787828828787806, 608787831138787806, 608787834438787806, 608787837738787806, 608787840048787806, 608787843348787806, 608787846648787806, 608787849948787806, 608787852258787806, 608787855558787806]
     7        [748867469964768847, 748867472274768847, 748867479974768847, 748867482284768847, 748867489984768847, 748867492294768847, 748867499994768847, 748867503305768847, 748867513315768847, 748867523325768847]
     8        [808696959959696808, 808696960069696808, 808696961169696808, 808696962269696808, 808696963369696808, 808696964469696808, 808696965569696808, 808696966669696808, 808696967769696808, 808696968869696808]
     9        [968787702207787869, 968787711117787869, 968787720027787869, 968787729927787869, 968787738837787869, 968787747747787869, 968787756657787869, 968787765567787869, 968787774477787869, 968787783387787869]
1770.411549 seconds (13.02 G allocations: 443.799 GiB, 40.12% gc time)

Nim[edit]

Translation of: Crystal
Forms palindromes using number<->string conversions.
import strutils  # for number input
import times # for timing code execution
import unicode # for reversed
 
proc palindromicgapfuls(digit, count, keep: int): seq[uint64] =
var palcnt = 0 # count of gapful palindromes
let to_skip = count - keep # count of unwanted values to skip
var gapfuls = newSeq[uint64]() # array of palindromic gapfuls
let nn = digit * 11 # digit gapful divisor: 11, 22,...88, 99
var (power, base, basep) = (1, 1, 0)
while true:
if (power.inc; power and 1) == 0: base = base * 10
var base11 = base * 11 # value of middle two digits positions: 110..
var this_lo = base * digit # starting half for this digit: 10.. to 90..
var next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
while this_lo < next_lo - 1:
var (palindrome, palindrome_base, left_half) = (0'u64, 0'u64, this_lo.intToStr)
let right_half = left_half.reversed
if (power and 1) == 1: basep = base11; palindrome_base = (left_half & right_half).parseUInt
else: basep = base; left_half.removeSuffix("0"); palindrome_base = (left_half & right_half).parseUInt
for i in 0..9:
palindrome = palindrome_base + (basep * i).uint
if (palindrome mod nn.uint) == 0:
if palcnt < to_skip: (palcnt += 1; continue)
gapfuls.add(palindrome)
if gapfuls.len == keep: return gapfuls
this_lo += 10
 
let start = epochTime()
 
var (count, keep) = (20, 20)
echo("First 20 palindromic gapful numbers ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (100, 15)
echo("\nLast 15 of first 100 palindromic gapful numbers ending in:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (1_000, 10)
echo("\nLast 10 of first 1000 palindromic gapful numbers ending in:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (100_000, 1)
echo("\n100,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (1_000_000, 1)
echo("\n1,000,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (10_000_000, 1)
echo("\n10,000,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
echo (epochTime() - start)
System: I7-6700HQ, 3.5GHz, Linux Kernel 5.9.10, GCC 10.2.0, Nim 1.4.0
Compil: $ nim c --cc:gcc --d:danger palindromicgapfuls.nim
Run as: $ ./palindromicgapfuls
Time: 25.42800664901733 secs
Faster version performing number<->string conversions for palindromes.
import strutils  # for number input
import times # for timing code execution
import unicode # for reversed
 
proc palindromicgapfuls(digit, count, keep: int): seq[uint64] =
var palcnt = 0 # count of gapful palindromes
let to_skip = count - keep # count of unwanted values to skip
let nn = digit * 11 # digit gapful divisor: 11, 22,...88, 99
var (power, base, digit) = (1, 1u64, digit.uint64)
while true:
if (power.inc; power and 1) == 0: base *= 10
let base11 = base * 11 # value of middle two digits positions: 110..
let this_lo = base * digit # starting half for this digit: 10.. to 90..
let next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
for front_half in countup(this_lo, next_lo - 2, 10):
var
basep = base11
left_half = $front_half
let right_half = left_half.reversed
if (power and 1) == 0: basep = base; left_half.setLen left_half.len - 1
var palindrome = (left_half.add right_half; left_half).parseUInt.uint64
for _ in 0..9:
if palindrome mod nn.uint == 0: (palcnt.inc; if palcnt > to_skip: result.add palindrome)
palindrome += basep
if result.len >= keep: result.setLen(keep); return
 
let start = epochTime()
 
var (count, keep) = (20, 20)
echo("First 20 palindromic gapful numbers ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (100, 15)
echo("\nLast 15 of first 100 palindromic gapful numbers ending in:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (1_000, 10)
echo("\nLast 10 of first 1000 palindromic gapful numbers ending in:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (100_000, 1)
echo("\n100,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (1_000_000, 1)
echo("\n1,000,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (10_000_000, 1)
echo("\n10,000,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
echo (epochTime() - start)
System: I7-6700HQ, 3.5GHz, Linux Kernel 5.9.10, GCC 10.2.0, Nim 1.4.0
Compil: $ nim c -d:danger -d:lto --passC:-march=native palindromicgapfuls.nim
Run as: $ ./palindromicgapfuls
Time: 18.29568219184875 secs
Fastest: make palindromes directly numerically.
import times
 
proc make_palindrome(front_half: uint64, power: int): uint64 =
var res, front_half = front_half
if (power and 1) == 0: res = res div 10
while front_half > 0:
res = res * 10 + front_half mod 10
front_half = front_half div 10
res
 
proc palindromicgapfuls(digit, count, keep: int): seq[uint64] =
var (palcnt, digit) = (0, digit.uint64) # count of gapful palindromes
let to_skip = count - keep # count of unwanted values to skip
var gapfuls = newSeq[uint64]() # array of palindromic gapfuls
let dd = digit * 11 # digit gapful divisor: 11, 22,...88, 99
var (power, base) = (1, 1u64)
while true:
if (power.inc; power and 1) == 0: base = base * 10
var base11 = base * 11 # value of middle two digits positions: 110..
var this_lo = base * digit # starting half for this digit: 10.. to 90..
var next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
for front_half in countup(this_lo, next_lo - 2, 10):
let basep = if (power and 1) == 1: base11 else: base
var palindrome = make_palindrome(front_half, power)
for _ in 0..9:
if palindrome mod dd == 0: (palcnt.inc; if palcnt > to_skip: gapfuls.add(palindrome))
palindrome += basep
if gapfuls.len >= keep: return gapfuls[0..keep-1]
 
let start = epochTime()
 
var (count, keep) = (20, 20)
echo("First 20 palindromic gapful numbers ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (100, 15)
echo("\nLast 15 of first 100 palindromic gapful numbers ending in:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (1_000, 10)
echo("\nLast 10 of first 1000 palindromic gapful numbers ending in:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (100_000, 1)
echo("\n100,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (1_000_000, 1)
echo("\n1,000,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
(count, keep) = (10_000_000, 1)
echo("\n10,000,000th palindromic gapful number ending with:")
for digit in 1..9: echo(digit, " : ", palindromicgapfuls(digit, count, keep) )
 
echo (epochTime() - start)
System: I7-6700HQ, 3.5GHz, Linux Kernel 5.9.14, GCC 10.2.0, Nim 1.4.2
Compil: $ nim c --cc:gcc --d:danger palindromicgapfuls.nim
Run as: $ ./palindromicgapfuls
Time: 8.304139614105225 secs
Output:
First 20 palindromic gapful numbers ending with:
1 : @[121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
2 : @[242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
3 : @[363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
4 : @[484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
5 : @[5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
6 : @[6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
7 : @[7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
8 : @[8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
9 : @[9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]

Last 15 of first 100 palindromic gapful numbers ending in:
1 : @[165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
2 : @[265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
3 : @[30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
4 : @[4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
5 : @[565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
6 : @[60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
7 : @[72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
8 : @[80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
9 : @[95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]

Last 10 of first 1000 palindromic gapful numbers ending in:
1 : @[17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
2 : @[27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
3 : @[3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
4 : @[482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
5 : @[57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
6 : @[6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
7 : @[7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
8 : @[8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
9 : @[9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]

100,000th palindromic gapful number ending with:
1 : @[178788887871]
2 : @[278788887872]
3 : @[30878611687803]
4 : @[4833326233384]
5 : @[578789987875]
6 : @[60878611687806]
7 : @[74826144162847]
8 : @[80869688696808]
9 : @[96878077087869]

1,000,000th palindromic gapful number ending with:
1 : @[17878799787871]
2 : @[27878799787872]
3 : @[3087876666787803]
4 : @[483333272333384]
5 : @[57878799787875]
6 : @[6087876996787806]
7 : @[7487226666227847]
8 : @[8086969559696808]
9 : @[9687870990787869]

10,000,000th palindromic gapful number ending with:
1 : @[1787878888787871]
2 : @[2787878888787872]
3 : @[308787855558787803]
4 : @[48333332623333384]
5 : @[5787878998787875]
6 : @[608787855558787806]
7 : @[748867523325768847]
8 : @[808696968869696808]
9 : @[968787783387787869]

Pascal[edit]

Works with: Free Pascal

Creating palindromes by adding the right numbers one by one and the precalculated modulus of that numbers
So the numbers to check stays small in bitsize modsum ~16 Bit , n ~ 64 Bit.Dividing is therefore faster
Thinking about it, you don't need n = Uint64, only the value of the digit in that place is enough.
Of course this task has no relevance see digit 9 from 100,000 to 10,000,000

9 :   96878077087869
9 :  9687870990787869
9 : 968787783387787869
program PalinGap;
{$IFDEF FPC}
{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$CODEALIGN proc=16}{$ALIGN 16}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF}
//example 5 digits, digit d
// d000d
// +00100 10 -times delta[0] aka middle digit
//->d010d d020d d030d d040d d050d d060d d070d d080d d090d and
// d100d -> not palindromatic
//correct by -10x00100 and use the next delta for the next digitplaces
// d000d
//+ 01010 -> delta[1]
// d101d
// starting over again with delta[0] until delta[1] is used 10 times
type
tLimits = record
LoLmt,HiLmt:Uint64;
end;
const
base = 10;
 
var
delta : Array[0..9] of Uint64;
deltaBase: Array[0..9] of Uint64;
deltaMod : Array[0..9] of Uint32;
deltaModBase : Array[0..9] of Uint32;
 
IdxCnt : Array[0..9] of Uint32;
ModSum : UInt64;
dgtMod : UInt32;
 
procedure InitDelta(dgt:Byte;dgtCnt:Byte);
var
n : Uint64;
i,k,mid : NativeInt;
Begin
mid := (dgtCnt-1) DIV 2;
//create Add masks
For i := 0 to mid do
Begin
IF ODD(dgtCnt) then
//first 1,101,10001,1000001,100000001,10000000001
Begin
n := 1;
IF i> 0 then
Begin
For k := 1 to i do
n := n*(Base*Base);
inc(n);
end
end
Else //even
// first 11,1001,100001,10000001...
Begin
n := Base;
For k := 1 to i do
n := n*(Base*Base);
inc(n);
end;
// second move to the right place
// 1000000,10100000,10001000,10000010,100000001
dgtMod := (dgt*(Base+1));
For k := mid-1 DOWNTO i do
n := n*Base;
 
delta[i] := n;
deltaMod[i]:= n MOD dgtMod;
deltaBase[i] := base*n;
deltaModBase[i]:= (base*n) MOD dgtMod;
end;
//counter for digit position
For k := 0 to 9 do
IdxCnt[k] := Base;
end;
 
function NextPalin(n : Uint64;dgtcnt:NativeInt):Uint64;inline;
var
k,b: NativeInt;
begin
k := 0;
repeat
n := n+delta[k];
inc(ModSum,deltaMod[k]);
b := IdxCnt[k]-1;
IdxCnt[k]:= b;
IF b <> 0 then
break
else
Begin
n := n-deltaBase[k];
dec(ModSum,deltaModBase[k]);
IdxCnt[k]:= Base;
inc(k);
IF k = dgtCnt then
Begin
n := 0;
BREAK;
end;
end;
until false;
NextPalin := n;
end;
 
procedure OutPalinGap(lowLmt,HiLmt,dgt:NativeInt);
var
n : Uint64;
i,dgtcnt,mid :NativeInt;
begin
i:=1;
write(dgt,' :');
For dgtcnt := 3 to 20 do
Begin
mid := (dgtcnt-1) shr 1;
initDelta(dgt,dgtcnt);
n := dgt*delta[mid];// '10...01' -> 'd0...0d'
ModSum := n MOD dgtMod;
 
while (n <>0) AND (i< LowLmt) do
Begin
IF (ModSum MOD dgtMod) = 0 then
Begin
inc(i);
ModSum :=0;//reduce Modsum
end;
n := NextPalin(n,mid);
end;
 
while (n <>0) AND (i<= HiLmt) do
Begin
IF (ModSum MOD dgtMod) = 0 then
Begin
inc(i);
write(n:dgtcnt+1);
ModSum :=0;//reduce Modsum
end;
n := NextPalin(n,mid);
end;
IF (i > HiLmt) then
BREAK;
end;
writeln;
end;
 
var
dgt : NativeInt;
begin
writeln('palindromic gapful numbers from 1 to 20');
For dgt := 1 to 9 do
OutPalinGap(1,20,dgt);
writeln;
writeln('palindromic gapful numbers from 86 to 100');
For dgt := 1 to 9 do
OutPalinGap(86,100,dgt);
writeln;
writeln('palindromic gapful numbers from 991 to 1000');
For dgt := 1 to 9 do
OutPalinGap(991,1000,dgt);
writeln;
writeln('palindromic gapful number 100,000');
For dgt := 1 to 9 do
OutPalinGap(100000,100000,dgt);
writeln;
writeln('palindromic gapful number 1,000,000');
For dgt := 1 to 9 do
OutPalinGap(1000000,1000000,dgt);
writeln;
writeln('palindromic gapful number 10,000,000');
For dgt := 1 to 9 do
OutPalinGap(10000000,10000000,dgt);
writeln;
end.
Output:

palindromic gapful numbers from 1 to 20 1 : 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591 2 : 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392 3 : 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333 4 : 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804 5 : 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895 6 : 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336 7 : 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087 8 : 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648 9 : 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

palindromic gapful numbers from 86 to 100 1 : 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971 2 : 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972 3 : 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703 4 : 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284 5 : 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975 6 : 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806 7 : 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927 8 : 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808 9 : 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

palindromic gapful numbers from 991 to 1000 1 : 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871 2 : 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872 3 : 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803 4 : 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384 5 : 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875 6 : 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806 7 : 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547 8 : 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808 9 : 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

palindromic gapful number 100,000 1 : 178788887871 2 : 278788887872 3 : 30878611687803 4 : 4833326233384 5 : 578789987875 6 : 60878611687806 7 : 74826144162847 8 : 80869688696808 9 : 96878077087869

palindromic gapful number 1,000,000 1 : 17878799787871 2 : 27878799787872 3 : 3087876666787803 4 : 483333272333384 5 : 57878799787875 6 : 6087876996787806 7 : 7487226666227847 8 : 8086969559696808 9 : 9687870990787869

palindromic gapful number 10,000,000 1 : 1787878888787871 2 : 2787878888787872 3 : 308787855558787803 4 : 48333332623333384 5 : 5787878998787875 6 : 608787855558787806 7 : 748867523325768847 8 : 808696968869696808 9 : 968787783387787869

real 0m4,503s

Perl[edit]

Minor (and inefficient) tweak on the Gapful numbers task.

use strict;
use warnings;
use feature 'say';
 
use constant Inf => 1e10;
 
sub is_p_gapful {
my($d,$n) = @_;
return '' unless 0 == $n % 11;
my @digits = split //, $n;
$d eq $digits[0] and (0 == $n % ($digits[0].$digits[-1])) and $n eq join '', reverse @digits;
}
 
for ([1, 20], [86, 15]) {
my($offset, $count) = @$_;
say "Palindromic gapful numbers starting at $offset:";
for my $d ('1'..'9') {
my $n = 0; my $out = "$d: ";
$out .= do { $n+1 < $count+$offset ? (is_p_gapful($d,$_) and ++$n and $n >= $offset and "$_ ") : last } for 100 .. Inf;
say $out
}
say ''
}
Output:
Palindromic gapful numbers starting at 1:
1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

Palindromic gapful numbers starting at 86:
1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284
5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

Phix[edit]

Translation of: Go

Translation of Go, but trimmed back to bare minimum: you should not expect this to fare particularly well at the 10_000_000-level against the likes of Go/Pascal, though it should fare reasonably well against lesser beings... The version below beats 'em all, though, for now.

function reverse_n(atom s)
atom e = 0
while s>0 do
e = e*10 + remainder(s,10)
s = floor(s/10)
end while
return e
end function
 
constant mx = 1000,
data = {{1, 20, "%7d "}, {86, 100, "%8d "}, {991, 1000, "%10d "}}
 
include builtins\ordinal.e
 
procedure main()
sequence results = repeat(repeat({},9),mx)
for d=1 to 9 do -- (the start/end digit)
integer count = 0, pow = 1, fl = d*11
for nd=3 to 15 do -- (number of digits, usually quits early)
-- (obvs. 64-bit phix is fine with 19 digits, but 32-bit ain't)
bool odd = (remainder(nd,2)==1)
for s=d*pow to (d+1)*pow-1 do -- (eg 300 to 399)
integer e = reverse_n(s)
for m=0 to iff(odd?9:0) do -- (1 or 10 iterations)
atom p = e + iff(odd ? s*pow*100+m*pow*10
 : s*pow*10)
if remainder(p,fl)==0 then -- gapful!
count += 1
results[count][d] = p
-- (see? goto /is/ sometimes useful :-))
if count==mx then #ilASM{jmp :outer} end if
end if
end for
end for
if odd then pow *= 10 end if
end for
if count<mx then ?9/0 end if -- oh dear...
#ilASM{::outer}
end for
 
for i=1 to length(data) do
{integer s, integer e, string fmt} = data[i]
printf(1,"%,d%s to %,d%s palindromic gapful numbers (> 100) ending with:\n", {s,ord(s),e,ord(e)})
for d=1 to 9 do
printf(1,"%d: ",d)
for j=s to e do
printf(1,fmt,results[j][d])
end for
printf(1,"\n")
end for
printf(1,"\n")
end for
end procedure
main()
Output:
1st to 20th palindromic gapful numbers (> 100) ending with:
1:     121    1001    1111    1221    1331    1441    1551    1661    1771    1881    1991   10901   11011   12221   13431   14641   15851   17171   18381   19591
2:     242    2002    2112    2222    2332    2442    2552    2662    2772    2882    2992   20702   21912   22022   23232   24442   25652   26862   28182   29392
3:     363    3003    3333    3663    3993   31713   33033   36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333
4:     484    4004    4224    4444    4664    4884   40304   42724   44044   46464   48884  400004  401104  402204  403304  404404  405504  406604  407704  408804
5:    5005    5115    5225    5335    5445    5555    5665    5775    5885    5995   50105   51315   52525   53735   54945   55055   56265   57475   58685   59895
6:    6006    6336    6666    6996   61116   64746   66066   69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336
7:    7007    7777   77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087
8:    8008    8448    8888   80608   86768   88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648
9:    9009    9999   94149   99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999 9459549 9508059 9557559 9606069

86th to 100th palindromic gapful numbers (> 100) ending with:
1:   165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971
2:   265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4:  4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284
5:   565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

991st to 1,000th palindromic gapful numbers (> 100) ending with:
1:   17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871
2:   27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4:  482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384
5:   57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

Ludicrously fast to 10,000,000,000,000,000,000th[edit]

Astonishingly this is all done with standard precision numbers, < 253. You realise this is like ten thousand times the square of the previous limits, and still far faster.
I will credit Self_numbers#AppleScript and the comment by Nigel Galloway on the talk page for ideas that inspired me.

-- demo/rosetta/Palindromic_gapful_numbers.exw
--
-- A palindrome such as 9459549 can be constructed/broken down into
-- 9000009
-- 400040
-- 50500
-- 9000
--
-- Further, 9459549 rem 99 is the same as
-- (the sum of rem 99 on all of those pieces) rem 99
--
-- Finding eg 400040 rem 99 can also be simplified, it is of course the
-- same as (400000 rem 99 + 40 rem 99) rem 99, and further 40 rem 99
-- is the same as (4 rem 99[already known])*10 rem 99 [smaller nos].
--
-- Also, when filling a "hole", such as the final 9, we find
-- v
-- 9450549 rem 99 = 9
-- 9451549 rem 99 = 19,
-- 9452549 rem 99 = 29,
-- 9453549 rem 99 = 39,
-- 9454549 rem 99 = 49,
-- 9455549 rem 99 = 59,
-- 9456549 rem 99 = 69,
-- 9457549 rem 99 = 79,
-- 9458549 rem 99 = 89, and
-- 9459549 rem 99 = 0,
-- ^
-- in this case only the '9' fits.
--
-- But actually we can predict what will fit from the partial sum of
-- prior pieces rem 99, ie 9000009..50500, and the same can be said
-- when filling the 505-sized hole - what will "fit" depends not on
-- what the "outer" actually are, but what their sum rem 99 is, and
-- likewise for larger and larger holes.
-- If we later find ourselves looking at the same size hole, with
-- the same outer rem and the same rem 99 outmost requirement, we
-- would know instantly how many things are going to fit.
-- True, keeping full lists as the holes got bigger would probably
-- consume memory almost as fast as an SR-71, but a single count,
-- albeit one keyed on 4 conditions, we can cope. It turns out that
-- even by the 10^19th scan, we only hit 24,791 variations anyway.
-- As we stumble across larger and larger holes, what we learn can
-- be used to skip more and more similar, such that finding the
-- ten millionth item is almost as fast as the first millionth, as
-- opposed to the times 10 that you'd normally expect.
--
-- Note that if I stumble across a hole that will fit more than I'm
-- prepared to fully skip, I start going through things one-by-one,
-- but that's ok because many smaller but still quite big holes
-- will probably be skipped.
--
 
sequence cache = columnize(repeat(columnize({tagset(9)}),9))
-- ie {{{1}, {1}, {1}, {1}, {1}, {1}, {1}, {1}, {1}},
-- {{2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}, {2}},
-- {{3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}, {3}},
-- {{4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}, {4}},
-- {{5}, {5}, {5}, {5}, {5}, {5}, {5}, {5}, {5}},
-- {{6}, {6}, {6}, {6}, {6}, {6}, {6}, {6}, {6}},
-- {{7}, {7}, {7}, {7}, {7}, {7}, {7}, {7}, {7}},
-- {{8}, {8}, {8}, {8}, {8}, {8}, {8}, {8}, {8}},
-- {{9}, {9}, {9}, {9}, {9}, {9}, {9}, {9}, {9}}}
-- aka 1 rem 11 .. 1 rem 99 are all 1,
-- .. 9 rem 11 .. 9 rem 99 are all 9.
-- each gets extended with 10 rem 11 .. 10 rem 99,
-- 100 rem 11 .. 100 rem 99,
-- ... .. 900 rem 99, etc.
-- (not really worth trying to take advantage of any cycles
-- that might appear in such a relatively small table, as
-- it will be at most (on 64-bit) 9 * 9 * 42.)
--
function rmdr(integer digit, gap, pow, n)
--
-- digit is the outer 0..9 (obvs 0 always yields 0),
-- gap is zeroes between (-1,0,1,2,.. for eg 1,11,101,1001),
-- pow is trailing zeros (0,1,2,.. for eg 101,1010,10100),
-- n is 1..9 for 11..99
-- eg rmdr(4,3,2,1) yields remainder(4000400,11), but
-- that === remainder(remainder(4000000,11)+
-- remainder( 400,11),11), and
-- if k = remainder(4*10^(m-1),11) [already known] then
-- remainder(4*10^m,11) === remainder(k*10,11), so
-- we only need to keep a small table for each [digit][n].
-- Thus we avoid maths on 10^42-size numbers/needing gmp.
--
if digit=0 then return 0 end if
integer nn = n*11
while length(cache[digit][n])<gap+pow+2 do
cache[digit][n] &= remainder(cache[digit][n][$]*10,nn)
end while
integer g = iff(gap=-1?0:cache[digit][n][gap+pow+2])
return remainder(g+cache[digit][n][pow+1],nn)
end function
 
integer skipd = new_dict()
 
function palindromicgapfuls(sequence pals, string lhs, atom palcount, to_skip, count, integer l, r, p, dd)
--
-- pals: results (passing it up grants it automatic pass-by-reference status, which may help speedwise)
-- lhs: eg "945" of a potential 9459549 result
-- palcount, to_skip, count: self explanatory (aka got/ignore/target)
-- l: length of inner to be filled in
-- r: remainder of outer, eg remainder(9400049,11), built from rmdr()
-- p: left shift (should in fact always equal length(lhs), I think)
-- dd: outermost 1..9 (despite the name, it's a single digit)
--
sequence key = {l,r,p,dd}
integer node = getd_index(key,skipd)
atom skip = iff(node==null?count:getd_by_index(node,skipd))
if node!=null and (palcount+skip)<to_skip then
palcount += skip
else
skip = 0
for d=0 to 9 do
integer r2 = remainder(r+rmdr(d,l-2,p,dd),dd*11)
if l<=2 then
if r2=0 then
palcount += 1
if palcount<=to_skip then
skip += 1
else
pals = append(pals,lhs&repeat(d+'0',l)&reverse(lhs))
end if
end if
else
{pals,palcount,atom skipn} = palindromicgapfuls(pals, lhs&(d+'0'), palcount, to_skip, count, l-2, r2, p+1, dd)
skip += skipn
end if
if palcount==count then exit end if
end for
if palcount<to_skip then setd(key,skip,skipd) end if
end if
return {pals,palcount,skip}
end function
 
function collect(integer digit, atom count, keep)
atom to_skip = count - keep,
palcount = 0, l = 3
sequence pals = {}
string lhs = ""&(digit+'0') -- ie "1" or "2" .. or "9"
while palcount < count do
integer r = rmdr(digit,l-2,0,digit)
{pals,palcount} = palindromicgapfuls(pals,lhs,palcount,to_skip,count,l-2,r,1,digit)
l += 1
end while
return pals
end function
 
constant tests = {{20,20,1},{100,15,1},{1000,10,1},{10_000,5,1},
{100_000,1,1},{1_000_000,1,1},{10_000_000,1,1},
{100_000_000,1,9},{1000_000_000,1,9},{10_000_000_000,1,9},
{100_000_000_000,1,9},{1000_000_000_000,1,9},
{10_000_000_000_000,1,9},{100_000_000_000_000,1,9},
{1000_000_000_000_000,1,9},
{10_000_000_000_000_000_000,1,9}} -- 64 bit only
-- (any further and you'd need mpfr just to hold counts)
 
atom t0 = time(), count, keep, start
for i=1 to length(tests)-(machine_bits()!=64) do
{count, keep, start} = tests[i]
atom from = count-keep+1
string r = iff(count==keep?sprintf("First %d",{count}):
iff(keep>1?sprintf("Last %d of first %,d",{keep,count})
 :sprintf("%,dth",{count}))),
s = iff(keep=1?"":"s")
printf(1,"%s palindromic gapful number%s ending with:\n", {r,s})
sequence tags = tagset(9,start),
res = apply(true,collect,{tags,count,keep})
string fmt = sprintf("%%%ds",max(apply(join(res,{}),length)))
for j=1 to length(res) do
printf(1,"%d: %s\n",{tags[j],join(apply(true,sprintf,{{fmt},res[j]})," ")})
end for
puts(1,"\n")
end for
printf(1,"Completed in %s\n",elapsed(time()-t0))
Output:
First 20 palindromic gapful numbers ending with:
1:     121    1001    1111    1221    1331    1441    1551    1661    1771    1881    1991   10901   11011   12221   13431   14641   15851   17171   18381   19591
2:     242    2002    2112    2222    2332    2442    2552    2662    2772    2882    2992   20702   21912   22022   23232   24442   25652   26862   28182   29392
3:     363    3003    3333    3663    3993   31713   33033   36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333
4:     484    4004    4224    4444    4664    4884   40304   42724   44044   46464   48884  400004  401104  402204  403304  404404  405504  406604  407704  408804
5:    5005    5115    5225    5335    5445    5555    5665    5775    5885    5995   50105   51315   52525   53735   54945   55055   56265   57475   58685   59895
6:    6006    6336    6666    6996   61116   64746   66066   69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336
7:    7007    7777   77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087
8:    8008    8448    8888   80608   86768   88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648
9:    9009    9999   94149   99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999 9459549 9508059 9557559 9606069

Last 15 of first 100 palindromic gapful numbers ending with:
1:   165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971
2:   265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4:  4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284
5:   565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

Last 10 of first 1,000 palindromic gapful numbers ending with:
1:   17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871
2:   27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4:  482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384
5:   57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

Last 5 of first 10,000 palindromic gapful numbers ending with:
1:   1787557871   1787667871   1787777871   1787887871   1787997871
2:   2787557872   2787667872   2787777872   2787887872   2787997872
3: 308760067803 308763367803 308766667803 308769967803 308772277803
4:  48327872384  48329192384  48330303384  48331513384  48332723384
5:   5787557875   5787667875   5787777875   5787887875   5787997875
6: 608763367806 608766667806 608769967806 608772277806 608775577806
7: 746958859647 746961169647 746968869647 746971179647 746978879647
8: 808691196808 808692296808 808693396808 808694496808 808695596808
9: 968697796869 968706607869 968715517869 968724427869 968733337869

100,000th palindromic gapful number ending with:
1:   178788887871
2:   278788887872
3: 30878611687803
4:  4833326233384
5:   578789987875
6: 60878611687806
7: 74826144162847
8: 80869688696808
9: 96878077087869

1,000,000th palindromic gapful number ending with:
1:   17878799787871
2:   27878799787872
3: 3087876666787803
4:  483333272333384
5:   57878799787875
6: 6087876996787806
7: 7487226666227847
8: 8086969559696808
9: 9687870990787869

10,000,000th palindromic gapful number ending with:
1:   1787878888787871
2:   2787878888787872
3: 308787855558787803
4:  48333332623333384
5:   5787878998787875
6: 608787855558787806
7: 748867523325768847
8: 808696968869696808
9: 968787783387787869

100,000,000th palindromic gapful number ending with:
9: 96878786855868787869

1,000,000,000th palindromic gapful number ending with:
9: 9687878775995778787869

10,000,000,000th palindromic gapful number ending with:
9: 968787878661166878787869

100,000,000,000th palindromic gapful number ending with:
9: 96878787877355377878787869

1,000,000,000,000th palindromic gapful number ending with:
9: 9687878787863773687878787869

10,000,000,000,000th palindromic gapful number ending with:
9: 968787878787711117787878787869

100,000,000,000,000th palindromic gapful number ending with:
9: 96878787878786133168787878787869

1,000,000,000,000,000th palindromic gapful number ending with:
9: 9687878787878768778678787878787869

Completed in 0.7s

On 64bit you'll also get (in 0.8s)

10,000,000,000,000,000,000th palindromic gapful number ending with:
9: 968787878787878787639936787878787878787869

I might agree that the last entry does not feel very convincing. Depending on how much spare time you have, leave this running and it'll look better, uncomment the reset of count, to something that'll actually finish, and you'll start to believe. :-)

--count = 100000
while count do
count -= 1
 ?collect(9,count,1)
end while

Prolog[edit]

Works with: SWI Prolog
init_palindrome(Digit, p(10, Next, 0)):-
Next is Digit * 10 - 1.
 
next_palindrome(Digit, p(Power, Next, Even), p(Power1, Next2, Even1), Palindrome):-
Next1 is Next + 1,
(Next1 is Power * (Digit + 1) ->
(Even == 1 -> Power1 is Power * 10 ; Power1 = Power),
Next2 is Digit * Power1,
Even1 is 1 - Even
;
Power1 = Power,
Next2 = Next1,
Even1 = Even
),
(Even1 == 1 ->
X is 10 * Power1, Y = Next2
;
X = Power1, Y is Next2 // 10
),
reverse_number(Y, Z),
Palindrome is Next2 * X + Z.
 
reverse_number(N, R):-
reverse_number(N, 0, R).
 
reverse_number(0, Result, Result):-
!.
reverse_number(N, R, Result):-
R1 is R * 10 + N mod 10,
N1 is N // 10,
reverse_number(N1, R1, Result).
 
is_gapful(N):-
is_gapful(N, N).
 
is_gapful(N, M):-
M < 10,
!,
0 is N mod (N mod 10 + 10 * (M mod 10)).
is_gapful(N, M):-
M1 is M // 10,
is_gapful(N, M1).
 
find_palindromic_gapful_numbers(N, List):-
find_palindromic_gapful_numbers(N, 1, List).
 
find_palindromic_gapful_numbers(_, 10, []):-
!.
find_palindromic_gapful_numbers(N, Digit, [Numbers|Rest]):-
find_palindromic_gapful_numbers1(Digit, N, Numbers),
Next_digit is Digit + 1,
find_palindromic_gapful_numbers(N, Next_digit, Rest).
 
find_palindromic_gapful_numbers1(Digit, N, List):-
init_palindrome(Digit, P),
find_palindromic_gapful_numbers1(Digit, P, N, 0, List).
 
find_palindromic_gapful_numbers1(_, _, N, N, []):-
!.
find_palindromic_gapful_numbers1(Digit, P, N, Count, List):-
next_palindrome(Digit, P, P_next, Palindrome),
(is_gapful(Palindrome) ->
Count1 is Count + 1,
List = [Palindrome|Rest]
;
Count1 = Count,
List = Rest
),
find_palindromic_gapful_numbers1(Digit, P_next, N, Count1, Rest).
 
print_numbers(First, Last, Numbers):-
(First == 1 ->
writef("First %w palindromic gapful numbers ending in:\n", [Last])
;
Count is Last - First + 1,
writef("Last %w of first %w palindromic gapful numbers ending in:\n", [Count, Last])
),
print_numbers(First, Last, 1, Numbers),
nl.
 
print_numbers(_, _, 10, _):-
!.
print_numbers(First, Last, Digit, [N|Numbers]):-
writef("%w:", [Digit]),
print_numbers1(First, Last, 1, N),
Next_digit is Digit + 1,
print_numbers(First, Last, Next_digit, Numbers).
 
print_numbers1(_, Last, I, _):-
I > Last,
nl,
!.
print_numbers1(First, Last, I, [_|Numbers]):-
I < First,
!,
J is I + 1,
print_numbers1(First, Last, J, Numbers).
print_numbers1(First, Last, I, [N|Numbers]):-
writef(" %w", [N]),
J is I + 1,
print_numbers1(First, Last, J, Numbers).
 
main:-
find_palindromic_gapful_numbers(1000, Numbers),
print_numbers(1, 20, Numbers),
print_numbers(86, 100, Numbers),
print_numbers(991, 1000, Numbers).
Output:
First 20 palindromic gapful numbers ending in:
1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

Last 15 of first 100 palindromic gapful numbers ending in:
1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284
5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

Last 10 of first 1000 palindromic gapful numbers ending in:
1: 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871
2: 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4: 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384
5: 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

Python[edit]

Generators[edit]

Uses the technique of:

  1. generating all odd number of digits palindromes, in order.
  2. generating all even number of digits palindromes, in order.
  3. merge sorting those (unbounded) generators.
  4. then filtering the palindromes for gapful palindromic numbers >= 100

With the number generator the binning was straight-forward.
Runtime is short.

Note: Although this uses the idea of generating palindromes from the Geeks4geeks reference mentioned in the Factor entry, none of their code was used.

from itertools import count
from pprint import pformat
import re
import heapq
 
 
def pal_part_gen(odd=True):
for i in count(1):
fwd = str(i)
rev = fwd[::-1][1:] if odd else fwd[::-1]
yield int(fwd + rev)
 
def pal_ordered_gen():
yield from heapq.merge(pal_part_gen(odd=True), pal_part_gen(odd=False))
 
def is_gapful(x):
return (x % (int(str(x)[0]) * 10 + (x % 10)) == 0)
 
if __name__ == '__main__':
start = 100
for mx, last in [(20, 20), (100, 15), (1_000, 10)]:
print(f"\nLast {last} of the first {mx} binned-by-last digit "
f"gapful numbers >= {start}")
bin = {i: [] for i in range(1, 10)}
gen = (i for i in pal_ordered_gen() if i >= start and is_gapful(i))
while any(len(val) < mx for val in bin.values()):
g = next(gen)
val = bin[g % 10]
if len(val) < mx:
val.append(g)
b = {k:v[-last:] for k, v in bin.items()}
txt = pformat(b, width=220)
print('', re.sub(r"[{},\[\]]", '', txt))
Output:
Last 20 of the first 20 binned-by-last digit gapful numbers >= 100
 1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
 2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
 3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
 4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
 5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
 6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
 7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
 8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
 9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

Last 15 of the first 100 binned-by-last digit gapful numbers >= 100
 1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
 2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
 3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
 4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284
 5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
 6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
 7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
 8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
 9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

Last 10 of the first 1000 binned-by-last digit gapful numbers >= 100
 1: 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871
 2: 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872
 3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
 4: 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384
 5: 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875
 6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
 7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
 8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
 9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

Functional[edit]

'''Palindromic gapful numbers'''
 
from itertools import chain, count, islice, tee
from functools import reduce
 
 
# palindromicGapfuls :: () -> [Int]
def palindromicGapfuls():
'''A non-finite series of gapful palindromic numbers.
'''

def derived(digitsEven):
'''A palindrome of an even or odd number of digits,
obtained by appending either all or just the tail
of the reversed digits of n.
'''

def go(x):
s = str(x)
r = s[::-1]
return int((s + r) if digitsEven else (s + r[1:]))
return go
 
return filter(
lambda n: 0 == n % (int(str(n)[0]) * 10 + (n % 10)),
mergeInOrder(
map(derived(False), count(10))
)(map(derived(True), count(10)))
)
 
 
# --------------------------TESTS--------------------------
# main :: IO ()
def main():
'''Various samples of gapful palindromes grouped by final digit.'''
 
tpl = tee(palindromicGapfuls(), 9)
 
# sample :: (String, Int, Int) -> String
def sample(label, dropped, taken):
return fTable(label)(compose(cons(' '), str))(
compose(unwords, map_(str))
)(
compose(
take(taken),
drop(dropped),
lambda i: filter(
lambda x: i == x % 10,
tpl[i - 1]
)
)
)(enumFromTo(1)(9))
 
print(
'\n\n'.join(map(lambda x: sample(*x), [
('First 20 samples of gapful palindromes ' +
'(> 100) by last digit:', 0, 20),
 
('Last 15 of first 100 gapful palindromes ' +
'(> 100) by last digit:', 65, 15),
 
('Last 10 of first 1000 gapful palindromes ' +
'(> 100) by last digit:', 890, 10)
]))
)
 
# ------------------------DISPLAY -------------------------
 
 
# fTable :: String -> (a -> String) ->
# (b -> String) -> (a -> b) -> [a] -> String
def fTable(s):
'''Heading -> x display function -> fx display function ->
f -> xs -> tabular string.
'''

def go(xShow, fxShow, f, xs):
ys = [xShow(x) for x in xs]
w = max(map(len, ys))
return s + '\n' + '\n'.join(map(
lambda x, y: y.rjust(w, ' ') + ': ' + fxShow(f(x)),
xs, ys
))
return lambda xShow: lambda fxShow: lambda f: lambda xs: go(
xShow, fxShow, f, xs
)
 
 
# ------------------------GENERIC--------------------------
 
# Just :: a -> Maybe a
def Just(x):
'''Constructor for an inhabited Maybe (option type) value.
Wrapper containing the result of a computation.
'''

return {'type': 'Maybe', 'Nothing': False, 'Just': x}
 
 
# Nothing :: Maybe a
def Nothing():
'''Constructor for an empty Maybe (option type) value.
Empty wrapper returned where a computation is not possible.
'''

return {'type': 'Maybe', 'Nothing': True}
 
 
# compose :: ((a -> a), ...) -> (a -> a)
def compose(*fs):
'''Composition, from right to left,
of a series of functions.
'''

def go(f, g):
return lambda x: f(g(x))
return reduce(go, fs, lambda x: x)
 
 
# cons :: a -> [a] -> [a]
def cons(x):
'''A list string or iterator constructed
from x as head, and xs as tail.
'''

return lambda xs: [x] + xs if (
isinstance(xs, list)
) else x + xs if (
isinstance(xs, str)
) else chain([x], xs)
 
 
# drop :: Int -> [a] -> [a]
def drop(n):
'''The sublist of xs beginning at
(zero-based) index n.
'''

def go(xs):
take(n)(xs)
return xs
return go
 
 
# enumFromTo :: Int -> Int -> [Int]
def enumFromTo(m):
'''Enumeration of integer values [m..n]'''
def go(n):
return list(range(m, 1 + n))
return go
 
 
# map :: (a -> b) -> [a] -> [b]
def map_(f):
'''The list obtained by applying f
to each element of xs.
'''

return lambda xs: [f(x) for x in xs]
 
 
# mergeInOrder :: Gen [Int] -> Gen [Int] -> Gen [Int]
def mergeInOrder(ga):
'''An ordered, non-finite, stream of integers
obtained by merging two other such streams.
'''

def go(ma, mb):
a = ma
b = mb
while not a['Nothing'] and not b['Nothing']:
(a1, a2) = a['Just']
(b1, b2) = b['Just']
if a1 < b1:
yield a1
a = uncons(a2)
else:
yield b1
b = uncons(b2)
 
return lambda gb: go(uncons(ga), uncons(gb))
 
 
# take :: Int -> [a] -> [a]
def take(n):
'''The prefix of xs of length n,
or xs itself if n > length xs.
'''

return lambda xs: list(islice(xs, n))
 
 
# uncons :: [a] -> Maybe (a, [a])
def uncons(xs):
'''The deconstruction of a non-empty list
(or generator stream) into two parts:
a head value, and the remaining values.
'''

nxt = take(1)(xs)
return Just((nxt[0], xs)) if nxt else Nothing()
 
 
# unwords :: [String] -> String
def unwords(xs):
'''A space-separated string derived
from a list of words.
'''

return ' '.join(xs)
 
 
# MAIN ---
if __name__ == '__main__':
main()
Output:
First 20 samples of gapful palindromes (> 100) by last digit:
 1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
 2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
 3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
 4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
 5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
 6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
 7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
 8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
 9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

Last 15 of first 100 gapful palindromes (> 100) by last digit:
 1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
 2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
 3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
 4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284
 5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
 6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
 7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
 8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
 9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

Last 10 of first 1000 gapful palindromes (> 100) by last digit:
 1: 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871
 2: 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872
 3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
 4: 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384
 5: 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875
 6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
 7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
 8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
 9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

Raku[edit]

(formerly Perl 6)

Works with: Rakudo version 2019.07.1
constant @digits = '0','1','2','3','4','5','6','7','8','9';
 
# Infinite lazy iterator to generate palindromic "gap" numbers
my @npal = flat [ @digits ], [ '00','11','22','33','44','55','66','77','88','99' ],
{
sink @^previous, @^penultimate;
[ flat @digits.map: -> \digit { @penultimate.map: digit ~ * ~ digit } ]
}*;
 
# Individual lazy palindromic gapful number iterators for each start/end digit
my @gappal = (1..9).map: -> \digit {
my \divisor = digit + 10 * digit;
@npal.hyper.map: -> \this { next unless (my \test = digit ~ this ~ digit) %% divisor; test }
}
 
# Display
( "(Required) First 20 gapful palindromes:", ^20, 7
,"\n(Required) 86th through 100th:", 85..99, 8
,"\n(Optional) 991st through 1,000th:", 990..999, 10
,"\n(Extra stretchy) 9,995th through 10,000th:", 9994..9999, 12
,"\n(Meh) 100,000th:", 99999, 14
).hyper(:1batch).map: -> $caption, $range, $fmt {
my $now = now;
say $caption;
put "$_: " ~ @gappal[$_-1][|$range].fmt("%{$fmt}s") for 1..9;
say round( now - $now, .001 ), " seconds";
}
Output:
(Required) First 20 gapful palindromes:
1:     121    1001    1111    1221    1331    1441    1551    1661    1771    1881    1991   10901   11011   12221   13431   14641   15851   17171   18381   19591
2:     242    2002    2112    2222    2332    2442    2552    2662    2772    2882    2992   20702   21912   22022   23232   24442   25652   26862   28182   29392
3:     363    3003    3333    3663    3993   31713   33033   36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333
4:     484    4004    4224    4444    4664    4884   40304   42724   44044   46464   48884  400004  401104  402204  403304  404404  405504  406604  407704  408804
5:    5005    5115    5225    5335    5445    5555    5665    5775    5885    5995   50105   51315   52525   53735   54945   55055   56265   57475   58685   59895
6:    6006    6336    6666    6996   61116   64746   66066   69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336
7:    7007    7777   77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087
8:    8008    8448    8888   80608   86768   88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648
9:    9009    9999   94149   99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999 9459549 9508059 9557559 9606069
0.111 seconds

(Required) 86th through 100th:
1:   165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971
2:   265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4:  4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284
5:   565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569
0.046 seconds

(Optional) 991st through 1,000th:
1:   17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871
2:   27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4:  482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384
5:   57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869
0.282 seconds

(Extra stretchy) 9,995th through 10,000th:
1:   1787447871   1787557871   1787667871   1787777871   1787887871   1787997871
2:   2787447872   2787557872   2787667872   2787777872   2787887872   2787997872
3: 308757757803 308760067803 308763367803 308766667803 308769967803 308772277803
4:  48326662384  48327872384  48329192384  48330303384  48331513384  48332723384
5:   5787447875   5787557875   5787667875   5787777875   5787887875   5787997875
6: 608760067806 608763367806 608766667806 608769967806 608772277806 608775577806
7: 746951159647 746958859647 746961169647 746968869647 746971179647 746978879647
8: 808690096808 808691196808 808692296808 808693396808 808694496808 808695596808
9: 968688886869 968697796869 968706607869 968715517869 968724427869 968733337869
3.114 seconds

(Meh) 100,000th:
1:   178788887871
2:   278788887872
3: 30878611687803
4:  4833326233384
5:   578789987875
6: 60878611687806
7: 74826144162847
8: 80869688696808
9: 96878077087869
32.603 seconds

REXX[edit]

/*REXX program computes and displays palindromic gapful numbers, it also can show those */
/*─────────────────────── palindromic gapful numbers listed by their last decimal digit.*/
numeric digits 20 /*ensure enough decimal digits gapfuls.*/
parse arg palGaps /*obtain optional arguments from the CL*/
if palGaps='' then palGaps= 20 100@@15 1000@@10 /*Not specified? Then use the defaults*/
 
do until palGaps=''; parse var palGaps stuff palGaps; call palGap stuff
end /*until*/
exit 0 /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
palGap: procedure; parse arg n '@' sp "@" z; #= 0; if sp=='' then sp= 100
@ending= ' (ending in a specific digit) '; if z=='' then z= n
@which= ' last '; if z==n then @which= " first "
@palGap#Start= ' palindromic gapful numbers starting at: '
say center(@which z ' of ' n @palGap#Start sp" " @ending, 140, "═")
#.= 0 /*array of result counts for each digit*/
newSP= max(110, sp%11*11) /*calculate the new starting point. */
tot= n * 9 /*total # of results that are wanted. */
$.=; sum= 0 /*blank lists; digit results (so far).*/
do j=newSP by 11 until sum==tot /*loop 'til all digit counters filled. */
if reverse(j) \==j then iterate /*Not a palindrome? Then skip it.*/
parse var j a 2 '' -1 b /*obtain the first and last dec. digit.*/
if #.b ==n then iterate /*Digit quota filled? Then skip it.*/
if j // (a||b) \==0 then iterate /*Not divisible by A||B? " " " */
sum= sum + 1; #.b= #.b + 1 /*bump the sum counter & digit counter.*/
$.b= $.b j /*append J to the correct list. */
end /*j*/
/* [↓] just show the last Z numbers.*/
do k=1 for 9; say k':' strip( subword($.k, 1 + n - z) )
end /*k*/; say
return
output   when using the internal default inputs:

(Shown at   5/6   size.)

═════════════════════ first  20  of  20  palindromic gapful numbers starting at:  100    (ending in a specific digit) ══════════════════════
1: 121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
2: 242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
3: 363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
4: 484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
5: 5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
6: 6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
7: 7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
8: 8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
9: 9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069

═════════════════════ last  15  of  100  palindromic gapful numbers starting at:  100    (ending in a specific digit) ══════════════════════
1: 165561 166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971
2: 265562 266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4: 4473744 4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284
5: 565565 566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

═════════════════════ last  10  of  1000  palindromic gapful numbers starting at:  100    (ending in a specific digit) ═════════════════════
1: 17799771 17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871
2: 27799772 27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4: 482282284 482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384
5: 57800875 57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

Ring[edit]

 
load "stdlib.ring"
 
see "First 20 palindromic gapful numbers > 100 ending with each digit from 1 to 9:" + nl
 
limit = 9606069
 
for n = 1 to 9
sum = 0
for x = 101 to limit
flag = 0
strx = string(x)
xbegin = left(strx,1)
xend = right(strx,1)
xnew = number(xbegin+xend)
for y = 2 to ceil(x/2)+1
if ispalindrome(strx)
if y != xnew and x % y != 0
if x % xnew = 0 and number(xend) = n
flag = 1
else
flag = 0
exit
ok
ok
ok
next
if flag = 1
sum = sum + 1
if sum < 21
see "x = " + x + nl
else
exit
ok
ok
next
see nl
next
 
see "done..." + nl
 

Output:

First 20 palindromic gapful numbers > 100 ending with each digit from 1 to 9:
121  1001  1111  1221  1331  1441  1551  1661  1771  1881  1991  10901  11011  12221  13431  14641  15851  17171  18381  19591
242  2002  2112  2222  2332  2442  2552  2662  2772  2882  2992  20702  21912  22022  23232  24442  25652  26862  28182  29392
363  3003  3333  3663  3993  31713  33033  36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333
484  4004  4224  4444  4664  4884  40304  42724  44044  46464  48884  400004  401104  402204  403304  404404  405504  406604  407704  408804
5005  5115  5225  5335  5445  5555  5665  5775  5885  5995  50105  51315  52525  53735  54945  55055  56265  57475  58685  59895
6006  6336  6666  6996  61116  64746  66066  69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336
7007  7777  77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087
8008  8448  8888  80608  86768  88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648
9009  9999  94149  99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999  9459549  9508059  9557559  9606069

Ruby[edit]

Translation of: Crystal

Brute force and slow[edit]

def palindromesgapful(digit, pow)
r1 = digit * (10**pow + 1)
r2 = 10**pow * (digit + 1)
nn = digit * 11
(r1...r2).select { |i| n = i.to_s; n == n.reverse && i % nn == 0 }
end
 
def digitscount(digit, count)
pow = 2
nums = []
while nums.size < count
nums += palindromesgapful(digit, pow)
pow += 1
end
nums[0...count]
end
 
count = 20
puts "First 20 palindromic gapful numbers ending with:"
(1..9).each { |digit| print "#{digit} : #{digitscount(digit, count)}\n" }
 
count = 100
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
(1..9).each { |digit| print "#{digit} : #{digitscount(digit, count).last(15)}\n" }
 
count = 1000
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
(1..9).each { |digit| print "#{digit} : #{digitscount(digit, count).last(10)}\n" }

Orders of Magnitude Faster: Direct Generation of Numbers[edit]

Ruby is a dynamic language evaluated at runtime.
The code as implemented has been tested to produce optimum performance.

System: I7-6700HQ, 3.5 GHz, Linux Kernel 5.6.13
Run as: $ ruby palindromicgapfuls.rb

Optimized version, the ultimate fastest: Ruby 2.7.1 - 112.5 secs

def palindromicgapfuls(digit, count)
gapfuls = [] # array of palindromic gapfuls
nn = digit * 11 # digit gapful divisor: 11, 22,...88, 99
power = 1 # these two lines will work
while power += 1 # for all Ruby VMs|versions
#(2..).each do |power| # Ruby => 2.6; can replace above 2 lines
base = 10**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_i
10.times do
gapfuls << palindrome if palindrome % nn == 0
palindrome += base11
end
else
palindrome = (left_half.chop + right_half).to_i
10.times do
gapfuls << palindrome if palindrome % nn == 0
palindrome += base
end
end
return gapfuls[0...count] unless gapfuls.size < count
end
end
end
 
start = Time.now
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
puts (Time.now - start)

Compact version: Ruby-2.7.1 - 113.0 secs

def palindromicgapfuls(digit, count)
gapfuls = [] # array of palindromic gapfuls
nn = digit * 11 # digit gapful divisor: 11, 22,...88, 99
power = 1 # these two lines will work
while power += 1 # for all Ruby VMs|versions
#(2..).each do |power| # Ruby => 2.6; can replace above 2 lines
base = 10**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half, basep = front_half.to_s, base11; right_half = left_half.reverse
(basep = base; left_half = left_half.chop) if power.even?
palindrome = (left_half + right_half).to_i
10.times do
gapfuls << palindrome if palindrome % nn == 0
palindrome += basep
end
return gapfuls[0...count] unless gapfuls.size < count
end
end
end
 
start = Time.now
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count).last(keep)}" }
 
puts (Time.now - start)

Object Oriented implementation: Ruby 2.7.1 - 113.0 secs

class PalindromicGapfuls
include Enumerable
 
def initialize(digit)
@digit = digit
@nn = @digit * 11 # digit gapful divisor: 11, 22,...88, 99
end
 
def each
power = 1 # these two lines will work
while power += 1 # for all Ruby VMs|versions
#(2..).each do |power| # Ruby => 2.6; can replace above 2 lines
base = 10**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * @digit # starting half for this digit: 10.. to 90..
next_lo = base * (@digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_i
10.times do
yield palindrome if palindrome % @nn == 0
palindrome += base11
end
else
palindrome = (left_half.chop + right_half).to_i
10.times do
yield palindrome if palindrome % @nn == 0
palindrome += base
end
end
end
end
end
end
 
start = Time.now
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).first(count).last(keep)}" }
 
puts (Time.now - start)

Versions optimized for minimal memory use: Ruby 2.7.1 - 110.0 secs

def palindromicgapfuls(digit, count, keep)
palcnt = 0 # count of gapful palindromes
to_skip = count - keep # count of unwanted values to skip
gapfuls = [] # array of palindromic gapfuls
nn = digit * 11 # digit gapful divisor: 11, 22,...88, 99
power = 1 # these two lines will work
while power += 1 # for all Ruby VMs|versions
#(2..).each do |power| # Ruby => 2.6; can replace above 2 lines
base = 10**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_i
10.times do
(gapfuls << palindrome if (palcnt += 1) > to_skip) if palindrome % nn == 0
palindrome += base11
end
else
palindrome = (left_half.chop + right_half).to_i
10.times do
(gapfuls << palindrome if (palcnt += 1) > to_skip) if palindrome % nn == 0
palindrome += base
end
end
return gapfuls[0...keep] unless gapfuls.size < keep
end
end
end
 
start = Time.now
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
puts (Time.now - start)

Compact version optimized for minimal memory use: Ruby 2.7.1 - 111.5 secs

def palindromicgapfuls(digit, count, keep)
palcnt = 0 # count of gapful palindromes
to_skip = count - keep # count of unwanted values to skip
gapfuls = [] # array of palindromic gapfuls
nn = digit * 11 # digit gapful divisor: 11, 22,...88, 99
power = 1 # these two lines will work
while power += 1 # for all Ruby VMs|versions
#(2..).each do |power| # Ruby => 2.6; can replace above 2 lines
base = 10 ** (power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * digit # starting half for this digit: 10.. to 90..
next_lo = base * (digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half, basep = front_half.to_s, base11; right_half = left_half.reverse
(basep = base; left_half = left_half.chop) if power.even?
palindrome = (left_half + right_half).to_i
10.times do
(gapfuls << palindrome if (palcnt += 1) > to_skip) if palindrome % nn == 0
palindrome += basep
end
return gapfuls[0...keep] unless gapfuls.size < keep
end
end
end
 
start = Time.now
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{palindromicgapfuls(digit, count, keep)}" }
 
puts (Time.now - start)

OOP version optimized for minimal memory use: Ruby 2.7.1 - 116.0 secs
It creates an output method that skips the unwanted values and only keeps/stores the desired ones.

class PalindromicGapfuls
include Enumerable
 
def initialize(digit)
@digit = digit
@nn = @digit * 11 # digit gapful divisor: 11, 22,...88, 99
end
 
def each
power = 1 # these two lines will work
while power += 1 # for all Ruby VMs|versions
#(2..).each do |power| # Ruby => 2.6; can replace above 2 lines
base = 10**(power >> 1) # value of middle digit position: 10..
base11 = base * 11 # value of middle two digits positions: 110..
this_lo = base * @digit # starting half for this digit: 10.. to 90..
next_lo = base * (@digit + 1) # starting half for next digit: 20.. to 100..
this_lo.step(to: next_lo - 1, by: 10) do |front_half| # d_00; d_10; d_20; ...
left_half = front_half.to_s; right_half = left_half.reverse
if power.odd?
palindrome = (left_half + right_half).to_i
10.times do
yield palindrome if palindrome % @nn == 0
palindrome += base11
end
else
palindrome = (left_half.chop + right_half).to_i
10.times do
yield palindrome if palindrome % @nn == 0
palindrome += base
end
end
end
end
end
 
# Optimized output method: only keep desired values.
def keep_from(count, keep)
to_skip = (count - keep)
kept = []
each_with_index do |value, i|
i < to_skip ? next : kept << value
return kept if kept.size == keep
end
end
end
 
start = Time.now
 
count, keep = 20, 20
puts "First 20 palindromic gapful numbers ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 100, 15
puts "\nLast 15 of first 100 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 1_000, 10
puts "\nLast 10 of first 1000 palindromic gapful numbers ending in:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 100_000, 1
puts "\n100,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 1_000_000, 1
puts "\n1,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
count, keep = 10_000_000, 1
puts "\n10,000,000th palindromic gapful number ending with:"
1.upto(9) { |digit| puts "#{digit} : #{PalindromicGapfuls.new(digit).keep_from(count, keep)}" }
 
puts (Time.now - start)
Output:
First 20 palindromic gapful numbers 100 ending with:
1 : [121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
2 : [242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
3 : [363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
4 : [484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
5 : [5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
6 : [6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
7 : [7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
8 : [8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
9 : [9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]

Last 15 of first 100 palindromic gapful numbers ending in:
1 : [165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
2 : [265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
3 : [30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
4 : [4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
5 : [565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
6 : [60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
7 : [72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
8 : [80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
9 : [95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]

Last 10 of first 1000 palindromic gapful numbers ending in:
1 : [17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
2 : [27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
3 : [3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
4 : [482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
5 : [57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
6 : [6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
7 : [7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
8 : [8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
9 : [9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]

100,000th palindromic gapful number ending with:
1 : [178788887871]
2 : [278788887872]
3 : [30878611687803]
4 : [4833326233384]
5 : [578789987875]
6 : [60878611687806]
7 : [74826144162847]
8 : [80869688696808]
9 : [96878077087869]

1,000,000th palindromic gapful number ending with:
1 : [17878799787871]
2 : [27878799787872]
3 : [3087876666787803]
4 : [483333272333384]
5 : [57878799787875]
6 : [6087876996787806]
7 : [7487226666227847]
8 : [8086969559696808]
9 : [9687870990787869]

10,000,000th palindromic gapful number ending with:
1 : [1787878888787871]
2 : [2787878888787872]
3 : [308787855558787803]
4 : [48333332623333384]
5 : [5787878998787875]
6 : [608787855558787806]
7 : [748867523325768847]
8 : [808696968869696808]
9 : [968787783387787869]

Translation of F#[edit]

Translation of: F#

0.186s on Intel(R) Core(TM) i5-1035G1 CPU @ 1.00GHz

 
class PalNo
def initialize(set)
@set, @l=set, 3
end
def fN(n)
return [0,1,2,3,4,5,6,7,8,9] if n==1
return [0,11,22,33,44,55,66,77,88,99] if n==2
a=[]; [0,1,2,3,4,5,6,7,8,9].product(fN(n-2)).each{|g| a.push(g[0]*10**(n-1)+g[0]+10*g[1])}; return a
end
def each
while true do fN(@l-2).each{|g| [email protected]*10**(@l-1)+@set+10*g; yield a if a%(11*@set)==0}; @l+=1 end
end
end
 
for n in 1..9 do palNo=PalNo.new(n); g=1; palNo.each{|n| print "#{n} "; g+=1; break unless g<21}; puts "" end; puts "####"
for n in 1..9 do palNo=PalNo.new(n); g=1; palNo.each{|n| print "#{n} " if g>85; g+=1; break unless g<101}; puts "" end; puts "####"
for n in 1..9 do palNo=PalNo.new(n); g=1; palNo.each{|n| print "#{n} " if g>990; g+=1; break unless g<1001}; puts "" end; puts "####"
 
Output:
121 1001 1111 1221 1331 1441 1551 1661 1771 1881 1991 10901 11011 12221 13431 14641 15851 17171 18381 19591
242 2002 2112 2222 2332 2442 2552 2662 2772 2882 2992 20702 21912 22022 23232 24442 25652 26862 28182 29392
363 3003 3333 3663 3993 31713 33033 36663 300003 303303 306603 309903 312213 315513 318813 321123 324423 327723 330033 333333
484 4004 4224 4444 4664 4884 40304 42724 44044 46464 48884 400004 401104 402204 403304 404404 405504 406604 407704 408804
5005 5115 5225 5335 5445 5555 5665 5775 5885 5995 50105 51315 52525 53735 54945 55055 56265 57475 58685 59895
6006 6336 6666 6996 61116 64746 66066 69696 600006 603306 606606 609906 612216 615516 618816 621126 624426 627726 630036 633336
7007 7777 77077 700007 707707 710017 717717 720027 727727 730037 737737 740047 747747 750057 757757 760067 767767 770077 777777 780087
8008 8448 8888 80608 86768 88088 800008 802208 804408 806608 808808 821128 823328 825528 827728 829928 840048 842248 844448 846648
9009 9999 94149 99099 900009 909909 918819 927729 936639 945549 954459 963369 972279 981189 990099 999999 9459549 9508059 9557559 9606069
####
166661 167761 168861 169961 170071 171171 172271 173371 174471 175571 176671 177771 178871 179971 180081
266662 267762 268862 269962 270072 271172 272272 273372 274472 275572 276672 277772 278872 279972 280082
30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703 30811803
4485844 4497944 4607064 4619164 4620264 4632364 4644464 4656564 4668664 4681864 4693964 4803084 4815184 4827284 4839384
566665 567765 568865 569965 570075 571175 572275 573375 574475 575575 576675 577775 578875 579975 580085
60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806 60844806
72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927 73033037
80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808 80866808
95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569 96666669
####
17800871 17811871 17822871 17833871 17844871 17855871 17866871 17877871 17888871 17899871
27800872 27811872 27822872 27833872 27844872 27855872 27866872 27877872 27888872 27899872
3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803 3087007803
482414284 482535284 482656284 482777284 482898284 482909284 483020384 483141384 483262384 483383384
57811875 57822875 57833875 57844875 57855875 57866875 57877875 57888875 57899875 57900975
6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806 6087007806
7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547 7457997547
8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808 8086996808
9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869 9684004869
####
Translation of: Ruby of F#


More user friendly|simpler implementation; max memory consumption hits ~50%.

System: I7-6700HQ, 3.5 GHz, 16GB, Linux Kernel 5.9.10, Ruby 2.7.2
Run as: $ ruby fsharp2ruby.rb
Best Time: 437.566515299 secs
class PalNo
def initialize(digit)
@digit, @l, @dd = digit, 3, 11*digit
end
def fN(n)
return [0,1,2,3,4,5,6,7,8,9] if n==1
return [0,11,22,33,44,55,66,77,88,99] if n==2
a=[]; [0,1,2,3,4,5,6,7,8,9].product(fN(n-2)).each{ |g0,g1| a << g0*10**(n-1)+g0+10*g1 }; return a
end
def show(count, keep)
to_skip, palcnt, pals = count - keep, 0, []
while palcnt < count
fN(@l-2).each{ |g| [email protected]*10**(@l-1)+@digit+10*g;
pals << pal if pal%(@dd)==0 && (palcnt += 1) > to_skip; break if palcnt - to_skip == keep }; @l+=1
end
print pals; puts
end
end
 
start = Time.now
 
(1..9).each { |digit| PalNo.new(digit).show(20, 20) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(100, 15) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(1000, 10) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(100_000, 1) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(1_000_000, 1) }; puts "####"
(1..9).each { |digit| PalNo.new(digit).show(10_000_000, 1) }; puts "####"
 
puts (Time.now - start)
Output:
[121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
[242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
[363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
[484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
[5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
[6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
[7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
[8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
[9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]
####
[165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
[265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
[30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
[4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
[565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
[60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
[72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
[80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
[95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]
####
[17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
[27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
[3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
[482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
[57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
[6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
[7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
[8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
[9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]
####
[178788887871]
[278788887872]
[30878611687803]
[4833326233384]
[578789987875]
[60878611687806]
[74826144162847]
[80869688696808]
[96878077087869]
####
[17878799787871]
[27878799787872]
[3087876666787803]
[483333272333384]
[57878799787875]
[6087876996787806]
[7487226666227847]
[8086969559696808]
[9687870990787869]
####
[1787878888787871]
[2787878888787872]
[308787855558787803]
[48333332623333384]
[5787878998787875]
[608787855558787806]
[748867523325768847]
[808696968869696808]
[968787783387787869]
####
437.566515299

Rust[edit]

Translation of: Crystal
This version uses number->string then string->number conversions to create palindromes.
fn palindromicgapfuls(digit: u64, count: u64, keep: usize) -> Vec<u64> {
let mut palcnt = 0u64; // count of gapful palindromes
let to_skip = count - keep as u64; // count of unwanted values to skip
let mut gapfuls: Vec<u64> = vec![]; // array of palindromic gapfuls
let nn = digit * 11; // digit gapful divisor: 11, 22,...88, 99
let (mut power, mut base) = (1, 1u64);
loop { power += 1;
if power & 1 == 0 { base *= 10 }; // value of middle digit position: 10..
let base11 = base * 11; // value of middle two digits positions: 110..
let this_lo = base * digit; // starting half for this digit: 10.. to 90..
let next_lo = base * (digit + 1); // starting half for next digit: 20.. to 100..
for front_half in (this_lo..next_lo-1).step_by(10) { // d_00; d_10; d_20; ...
let (mut left_half, mut basep) = (front_half.to_string(), 0);
let right_half = left_half.chars().rev().collect::<String>();
if power & 1 == 1 { basep = base11; left_half.push_str(&right_half) }
else { basep = base; left_half.pop(); left_half.push_str(&right_half) };
let mut palindrome = left_half.parse::<u64>().unwrap();
for _ in 0..10 {
if palindrome % nn == 0 { palcnt += 1; if palcnt > to_skip { gapfuls.push(palindrome) } };
palindrome += basep;
}
if gapfuls.len() >= keep { return gapfuls[0..keep].to_vec() };
}
}
}
 
fn main() {
let t = std::time::Instant::now();
 
let (count, keep) = (20, 20);
println!("First 20 palindromic gapful numbers ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (100, 15);
println!("\nLast 15 of first 100 palindromic gapful numbers ending in:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (1_000, 10);
println!("\nLast 10 of first 1000 palindromic gapful numbers ending in:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (100_000, 1);
println!("\n100,000th palindromic gapful number ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (1_000_000, 1);
println!("\n1,000,000th palindromic gapful number ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (10_000_000, 1);
println!("\n10,000,000th palindromic gapful number ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
println!("{:?}", t.elapsed())
}
System: I7-6700HQ, 3.5 GHz, Linux Kernel 5.9.10, Rust 1.48
Compil: $ rustc -C opt-level=3 -C target-cpu=native -C codegen-units=1 -C lto palindromicgapfuls.rs 
Run as: ./palindromicgapfuls
Time: 19.973894976s


This version creates palindromes numerically instead of using number<->string conversions.
About 2.5x faster.
fn palindromicgapfuls(digit: u64, count: u64, keep: usize) -> Vec<u64> {
let mut palcnt = 0u64; // count of gapful palindromes
let to_skip = count - keep as u64; // count of unwanted values to skip
let mut gapfuls: Vec<u64> = vec![]; // array of palindromic gapfuls
let nn = digit * 11; // digit gapful divisor: 11, 22,...88, 99
let (mut power, mut base) = (1, 1u64);
loop { power += 1;
if power & 1 == 0 { base *= 10 } // value of middle digit position: 10..
let base11 = base * 11; // value of middle two digits positions: 110..
let this_lo = base * digit; // starting half for this digit: 10.. to 90..
let next_lo = base * (digit + 1); // starting half for next digit: 20.. to 100..
for front_half in (this_lo..next_lo).step_by(10) { // d_00; d_10; d_20; ...
let basep = if power & 1 == 1 { base11 } else { base };
let mut palindrome = make_palindrome(front_half, power);
for _ in 0..10 {
if palindrome % nn == 0 { palcnt += 1; if palcnt > to_skip { gapfuls.push(palindrome) } };
palindrome += basep;
}
if gapfuls.len() >= keep { return gapfuls[0..keep].to_vec() };
} } }
 
fn make_palindrome(mut front_half: u64, power: u64) -> u64 {
let mut result = front_half;
if power & 1 == 0 { result /= 10; }
while front_half > 0 {
result *= 10;
result += front_half % 10;
front_half /= 10;
}
result
}
 
fn main() {
let t = std::time::Instant::now();
 
let (count, keep) = (20, 20);
println!("First 20 palindromic gapful numbers ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (100, 15);
println!("\nLast 15 of first 100 palindromic gapful numbers ending in:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (1_000, 10);
println!("\nLast 10 of first 1000 palindromic gapful numbers ending in:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (100_000, 1);
println!("\n100,000th palindromic gapful number ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (1_000_000, 1);
println!("\n1,000,000th palindromic gapful number ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
let (count, keep) = (10_000_000, 1);
println!("\n10,000,000th palindromic gapful number ending with:");
for digit in 1..10 { println!("{} : {:?}", digit, palindromicgapfuls(digit, count, keep)); }
 
println!("{:?}", t.elapsed())
}
System: I7-6700HQ, 3.5 GHz, Linux Kernel 5.9.10, Rust 1.48
Compil: $ rustc -C opt-level=3 -C target-cpu=native -C codegen-units=1 -C lto palindromicgapfuls.rs 
Run as: ./palindromicgapfuls
Time: 8.768842134s
Output:
First 20 palindromic gapful numbers 100 ending with:
1 : [121, 1001, 1111, 1221, 1331, 1441, 1551, 1661, 1771, 1881, 1991, 10901, 11011, 12221, 13431, 14641, 15851, 17171, 18381, 19591]
2 : [242, 2002, 2112, 2222, 2332, 2442, 2552, 2662, 2772, 2882, 2992, 20702, 21912, 22022, 23232, 24442, 25652, 26862, 28182, 29392]
3 : [363, 3003, 3333, 3663, 3993, 31713, 33033, 36663, 300003, 303303, 306603, 309903, 312213, 315513, 318813, 321123, 324423, 327723, 330033, 333333]
4 : [484, 4004, 4224, 4444, 4664, 4884, 40304, 42724, 44044, 46464, 48884, 400004, 401104, 402204, 403304, 404404, 405504, 406604, 407704, 408804]
5 : [5005, 5115, 5225, 5335, 5445, 5555, 5665, 5775, 5885, 5995, 50105, 51315, 52525, 53735, 54945, 55055, 56265, 57475, 58685, 59895]
6 : [6006, 6336, 6666, 6996, 61116, 64746, 66066, 69696, 600006, 603306, 606606, 609906, 612216, 615516, 618816, 621126, 624426, 627726, 630036, 633336]
7 : [7007, 7777, 77077, 700007, 707707, 710017, 717717, 720027, 727727, 730037, 737737, 740047, 747747, 750057, 757757, 760067, 767767, 770077, 777777, 780087]
8 : [8008, 8448, 8888, 80608, 86768, 88088, 800008, 802208, 804408, 806608, 808808, 821128, 823328, 825528, 827728, 829928, 840048, 842248, 844448, 846648]
9 : [9009, 9999, 94149, 99099, 900009, 909909, 918819, 927729, 936639, 945549, 954459, 963369, 972279, 981189, 990099, 999999, 9459549, 9508059, 9557559, 9606069]

Last 15 of first 100 palindromic gapful numbers ending in:
1 : [165561, 166661, 167761, 168861, 169961, 170071, 171171, 172271, 173371, 174471, 175571, 176671, 177771, 178871, 179971]
2 : [265562, 266662, 267762, 268862, 269962, 270072, 271172, 272272, 273372, 274472, 275572, 276672, 277772, 278872, 279972]
3 : [30366303, 30399303, 30422403, 30455403, 30488403, 30511503, 30544503, 30577503, 30600603, 30633603, 30666603, 30699603, 30722703, 30755703, 30788703]
4 : [4473744, 4485844, 4497944, 4607064, 4619164, 4620264, 4632364, 4644464, 4656564, 4668664, 4681864, 4693964, 4803084, 4815184, 4827284]
5 : [565565, 566665, 567765, 568865, 569965, 570075, 571175, 572275, 573375, 574475, 575575, 576675, 577775, 578875, 579975]
6 : [60399306, 60422406, 60455406, 60488406, 60511506, 60544506, 60577506, 60600606, 60633606, 60666606, 60699606, 60722706, 60755706, 60788706, 60811806]
7 : [72299227, 72322327, 72399327, 72422427, 72499427, 72522527, 72599527, 72622627, 72699627, 72722727, 72799727, 72822827, 72899827, 72922927, 72999927]
8 : [80611608, 80622608, 80633608, 80644608, 80655608, 80666608, 80677608, 80688608, 80699608, 80800808, 80811808, 80822808, 80833808, 80844808, 80855808]
9 : [95311359, 95400459, 95499459, 95588559, 95677659, 95766759, 95855859, 95944959, 96033069, 96122169, 96211269, 96300369, 96399369, 96488469, 96577569]

Last 10 of first 1000 palindromic gapful numbers ending in:
1 : [17799771, 17800871, 17811871, 17822871, 17833871, 17844871, 17855871, 17866871, 17877871, 17888871]
2 : [27799772, 27800872, 27811872, 27822872, 27833872, 27844872, 27855872, 27866872, 27877872, 27888872]
3 : [3084004803, 3084334803, 3084664803, 3084994803, 3085225803, 3085555803, 3085885803, 3086116803, 3086446803, 3086776803]
4 : [482282284, 482414284, 482535284, 482656284, 482777284, 482898284, 482909284, 483020384, 483141384, 483262384]
5 : [57800875, 57811875, 57822875, 57833875, 57844875, 57855875, 57866875, 57877875, 57888875, 57899875]
6 : [6084004806, 6084334806, 6084664806, 6084994806, 6085225806, 6085555806, 6085885806, 6086116806, 6086446806, 6086776806]
7 : [7452992547, 7453223547, 7453993547, 7454224547, 7454994547, 7455225547, 7455995547, 7456226547, 7456996547, 7457227547]
8 : [8085995808, 8086006808, 8086116808, 8086226808, 8086336808, 8086446808, 8086556808, 8086666808, 8086776808, 8086886808]
9 : [9675005769, 9675995769, 9676886769, 9677777769, 9678668769, 9679559769, 9680440869, 9681331869, 9682222869, 9683113869]

100,000th palindromic gapful number ending with:
1 : [178788887871]
2 : [278788887872]
3 : [30878611687803]
4 : [4833326233384]
5 : [578789987875]
6 : [60878611687806]
7 : [74826144162847]
8 : [80869688696808]
9 : [96878077087869]

1,000,000th palindromic gapful number ending with:
1 : [17878799787871]
2 : [27878799787872]
3 : [3087876666787803]
4 : [483333272333384]
5 : [57878799787875]
6 : [6087876996787806]
7 : [7487226666227847]
8 : [8086969559696808]
9 : [9687870990787869]

10,000,000th palindromic gapful number ending with:
1 : [1787878888787871]
2 : [2787878888787872]
3 : [308787855558787803]
4 : [48333332623333384]
5 : [5787878998787875]
6 : [608787855558787806]
7 : [748867523325768847]
8 : [808696968869696808]
9 : [968787783387787869]

Sidef[edit]

Inspired from the C++ and Raku entries.

class PalindromeGenerator (digit, base=10) {
 
has power = base
has after = (digit*power - 1)
has even = false
 
method next {
 
if (++after == power*(digit+1)) {
power *= base if even
after = digit*power
even.not!
}
 
even ? (after*power*base + reverse(after, base))
 : (after*power + reverse(after/base, base))
}
}
 
var task = [
"(Required) First 20 gapful palindromes:", { .first(20) }, 7,
,"\n(Required) 86th through 100th:", { .first(1e2).last(15) }, 8,
,"\n(Optional) 991st through 1,000th:", { .first(1e3).last(10) }, 10,
,"\n(Extra stretchy) 9,995th through 10,000th:", { .first(1e4).last(6) }, 12,
]
 
task.each_slice(3, {|title, f, w|
say title
for d in (1..9) {
var k = 11*d
var iter = PalindromeGenerator(d)
var arr = f(^Inf->lazy.map { iter.next }.grep {|n| k `divides` n })
say ("#{d}: ", arr.map{ "%*s" % (w, _) }.join(' '))
}
})
Output:
(Required) First 20 gapful palindromes:
1:     121    1001    1111    1221    1331    1441    1551    1661    1771    1881    1991   10901   11011   12221   13431   14641   15851   17171   18381   19591
2:     242    2002    2112    2222    2332    2442    2552    2662    2772    2882    2992   20702   21912   22022   23232   24442   25652   26862   28182   29392
3:     363    3003    3333    3663    3993   31713   33033   36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333
4:     484    4004    4224    4444    4664    4884   40304   42724   44044   46464   48884  400004  401104  402204  403304  404404  405504  406604  407704  408804
5:    5005    5115    5225    5335    5445    5555    5665    5775    5885    5995   50105   51315   52525   53735   54945   55055   56265   57475   58685   59895
6:    6006    6336    6666    6996   61116   64746   66066   69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336
7:    7007    7777   77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087
8:    8008    8448    8888   80608   86768   88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648
9:    9009    9999   94149   99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999 9459549 9508059 9557559 9606069

(Required) 86th through 100th:
1:   165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971
2:   265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703
4:  4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284
5:   565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569

(Optional) 991st through 1,000th:
1:   17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871
2:   27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803
4:  482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384
5:   57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869

(Extra stretchy) 9,995th through 10,000th:
1:   1787447871   1787557871   1787667871   1787777871   1787887871   1787997871
2:   2787447872   2787557872   2787667872   2787777872   2787887872   2787997872
3: 308757757803 308760067803 308763367803 308766667803 308769967803 308772277803
4:  48326662384  48327872384  48329192384  48330303384  48331513384  48332723384
5:   5787447875   5787557875   5787667875   5787777875   5787887875   5787997875
6: 608760067806 608763367806 608766667806 608769967806 608772277806 608775577806
7: 746951159647 746958859647 746961169647 746968869647 746971179647 746978879647
8: 808690096808 808691196808 808692296808 808693396808 808694496808 808695596808
9: 968688886869 968697796869 968706607869 968715517869 968724427869 968733337869

Wren[edit]

Translation of: Go
Library: Wren-fmt

Search limited to the first 100,000 palindromic gapful numbers as, beyond that, the numbers become too large (>= 2 ^ 53) to be accurately represented by Wren's Num type.

import "/fmt" for Conv, Fmt
 
var reverse = Fn.new { |s|
var e = 0
while (s > 0) {
e = e * 10 + (s % 10)
s = (s/10).floor
}
return e
}
 
var MAX = 100000
var data = [ [1, 20, 7], [86, 100, 8], [991, 1000, 10], [9995, 10000, 12], [99996, 100000, 14] ]
var results = {}
for (d in data) {
for (i in d[0]..d[1]) results[i] = List.filled(9, 0)
}
var p
for (d in 1..9) {
var next_d = false
var count = 0
var pow = 1
var fl = d * 11
for (nd in 3..19) {
var slim = (d + 1) * pow
for (s in d*pow...slim) {
var e = reverse.call(s)
var mlim = (nd%2 != 1) ? 1 : 10
for (m in 0...mlim) {
if (nd%2 == 0) {
p = s*pow*10 + e
} else {
p = s*pow*100 + m*pow*10 + e
}
if (p%fl == 0) {
count = count + 1
var rc = results[count]
if (rc != null) rc[d-1] = p
if (count == MAX) next_d = true
}
if (next_d) break
}
if (next_d) break
}
if (next_d) break
if (nd%2 == 1) pow = pow * 10
}
}
 
for (d in data) {
var s1 = Fmt.ordinalize(d[0])
var s2 = Fmt.ordinalize(d[1])
System.print("%(s1) to %(s2) palindromic gapful numbers (> 100) ending with:")
for (i in 1..9) {
System.write("%(i): ")
for (j in d[0]..d[1]) System.write("%(Fmt.d(d[2], results[j][i-1])) ")
System.print()
}
System.print()
}
Output:
1st to 20th palindromic gapful numbers (> 100) ending with:
1:     121    1001    1111    1221    1331    1441    1551    1661    1771    1881    1991   10901   11011   12221   13431   14641   15851   17171   18381   19591 
2:     242    2002    2112    2222    2332    2442    2552    2662    2772    2882    2992   20702   21912   22022   23232   24442   25652   26862   28182   29392 
3:     363    3003    3333    3663    3993   31713   33033   36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333 
4:     484    4004    4224    4444    4664    4884   40304   42724   44044   46464   48884  400004  401104  402204  403304  404404  405504  406604  407704  408804 
5:    5005    5115    5225    5335    5445    5555    5665    5775    5885    5995   50105   51315   52525   53735   54945   55055   56265   57475   58685   59895 
6:    6006    6336    6666    6996   61116   64746   66066   69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336 
7:    7007    7777   77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087 
8:    8008    8448    8888   80608   86768   88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648 
9:    9009    9999   94149   99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999 9459549 9508059 9557559 9606069 

86th to 100th palindromic gapful numbers (> 100) ending with:
1:   165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971 
2:   265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972 
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703 
4:  4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284 
5:   565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975 
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806 
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927 
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808 
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569 

991st to 1,000th palindromic gapful numbers (> 100) ending with:
1:   17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871 
2:   27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872 
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803 
4:  482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384 
5:   57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875 
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806 
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547 
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808 
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869 

9,995th to 10,000th palindromic gapful numbers (> 100) ending with:
1:   1787447871   1787557871   1787667871   1787777871   1787887871   1787997871 
2:   2787447872   2787557872   2787667872   2787777872   2787887872   2787997872 
3: 308757757803 308760067803 308763367803 308766667803 308769967803 308772277803 
4:  48326662384  48327872384  48329192384  48330303384  48331513384  48332723384 
5:   5787447875   5787557875   5787667875   5787777875   5787887875   5787997875 
6: 608760067806 608763367806 608766667806 608769967806 608772277806 608775577806 
7: 746951159647 746958859647 746961169647 746968869647 746971179647 746978879647 
8: 808690096808 808691196808 808692296808 808693396808 808694496808 808695596808 
9: 968688886869 968697796869 968706607869 968715517869 968724427869 968733337869 

99,996th to 100,000th palindromic gapful numbers (> 100) ending with:
1:   178784487871   178785587871   178786687871   178787787871   178788887871 
2:   278784487872   278785587872   278786687872   278787787872   278788887872 
3: 30878499487803 30878522587803 30878555587803 30878588587803 30878611687803 
4:  4833289823384  4833290923384  4833302033384  4833314133384  4833326233384 
5:   578785587875   578786687875   578787787875   578788887875   578789987875 
6: 60878499487806 60878522587806 60878555587806 60878588587806 60878611687806 
7: 74825733752847 74825833852847 74825933952847 74826044062847 74826144162847 
8: 80869644696808 80869655696808 80869666696808 80869677696808 80869688696808 
9: 96877711777869 96877800877869 96877899877869 96877988977869 96878077087869 

zkl[edit]

Using ideas from the Factor entry

// 10,True  --> 101,111,121,131,141,151,161,171,181,191,202, ..
// 10,False --> 1001,1111,1221,1331,1441,1551,1661,1771,1881,..
fcn createPalindromeW(start,oddLength){ //--> iterator
[start..].tweak('wrap(z){
p,n := z,z;
if(oddLength) n/=10;
while(n>0){ p,n = p*10 + (n%10), n/10; }
p
})
}
fcn palindromicGapfulW(endsWith){ //--> iterator
po,pe := createPalindromeW(10,True), createPalindromeW(10,False);
div:=endsWith*10 + endsWith;
Walker.zero().tweak('wrap{
p:=( if(pe.peek()<po.peek()) pe.next() else po.next() );
if(p%10==endsWith and (p%div)==0) p else Void.Skip
})
}
println("First 20 palindromic gapful numbers:");
[1..9].apply(palindromicGapfulW).apply("walk",20) : pgpp(_);
 
foreach N,sz in (T( T(100,15), T(1_000,10), )){
println("\nLast %d of %,d palindromic gapful numbers:".fmt(sz,N));
[1..9].apply('wrap(n){ palindromicGapfulW(n).drop(N-sz).walk(sz) }) : pgpp(_);
}
 
fcn pgpp(table){ // pretty print ( (numbers),(numbers) )
m,fmt := (0).max(table.apply((0).max)).numDigits, "%%%dd ".fmt(m).fmt;
foreach d,row in ([1..].zip(table)){ println(d,": ",row.pump(String,fmt)) }
}
Output:
First 20 palindromic gapful numbers:
1:     121    1001    1111    1221    1331    1441    1551    1661    1771    1881    1991   10901   11011   12221   13431   14641   15851   17171   18381   19591 
2:     242    2002    2112    2222    2332    2442    2552    2662    2772    2882    2992   20702   21912   22022   23232   24442   25652   26862   28182   29392 
3:     363    3003    3333    3663    3993   31713   33033   36663  300003  303303  306603  309903  312213  315513  318813  321123  324423  327723  330033  333333 
4:     484    4004    4224    4444    4664    4884   40304   42724   44044   46464   48884  400004  401104  402204  403304  404404  405504  406604  407704  408804 
5:    5005    5115    5225    5335    5445    5555    5665    5775    5885    5995   50105   51315   52525   53735   54945   55055   56265   57475   58685   59895 
6:    6006    6336    6666    6996   61116   64746   66066   69696  600006  603306  606606  609906  612216  615516  618816  621126  624426  627726  630036  633336 
7:    7007    7777   77077  700007  707707  710017  717717  720027  727727  730037  737737  740047  747747  750057  757757  760067  767767  770077  777777  780087 
8:    8008    8448    8888   80608   86768   88088  800008  802208  804408  806608  808808  821128  823328  825528  827728  829928  840048  842248  844448  846648 
9:    9009    9999   94149   99099  900009  909909  918819  927729  936639  945549  954459  963369  972279  981189  990099  999999 9459549 9508059 9557559 9606069 

Last 15 of 100 palindromic gapful numbers:
1:   165561   166661   167761   168861   169961   170071   171171   172271   173371   174471   175571   176671   177771   178871   179971 
2:   265562   266662   267762   268862   269962   270072   271172   272272   273372   274472   275572   276672   277772   278872   279972 
3: 30366303 30399303 30422403 30455403 30488403 30511503 30544503 30577503 30600603 30633603 30666603 30699603 30722703 30755703 30788703 
4:  4473744  4485844  4497944  4607064  4619164  4620264  4632364  4644464  4656564  4668664  4681864  4693964  4803084  4815184  4827284 
5:   565565   566665   567765   568865   569965   570075   571175   572275   573375   574475   575575   576675   577775   578875   579975 
6: 60399306 60422406 60455406 60488406 60511506 60544506 60577506 60600606 60633606 60666606 60699606 60722706 60755706 60788706 60811806 
7: 72299227 72322327 72399327 72422427 72499427 72522527 72599527 72622627 72699627 72722727 72799727 72822827 72899827 72922927 72999927 
8: 80611608 80622608 80633608 80644608 80655608 80666608 80677608 80688608 80699608 80800808 80811808 80822808 80833808 80844808 80855808 
9: 95311359 95400459 95499459 95588559 95677659 95766759 95855859 95944959 96033069 96122169 96211269 96300369 96399369 96488469 96577569 

Last 10 of 1,000 palindromic gapful numbers:
1:   17799771   17800871   17811871   17822871   17833871   17844871   17855871   17866871   17877871   17888871 
2:   27799772   27800872   27811872   27822872   27833872   27844872   27855872   27866872   27877872   27888872 
3: 3084004803 3084334803 3084664803 3084994803 3085225803 3085555803 3085885803 3086116803 3086446803 3086776803 
4:  482282284  482414284  482535284  482656284  482777284  482898284  482909284  483020384  483141384  483262384 
5:   57800875   57811875   57822875   57833875   57844875   57855875   57866875   57877875   57888875   57899875 
6: 6084004806 6084334806 6084664806 6084994806 6085225806 6085555806 6085885806 6086116806 6086446806 6086776806 
7: 7452992547 7453223547 7453993547 7454224547 7454994547 7455225547 7455995547 7456226547 7456996547 7457227547 
8: 8085995808 8086006808 8086116808 8086226808 8086336808 8086446808 8086556808 8086666808 8086776808 8086886808 
9: 9675005769 9675995769 9676886769 9677777769 9678668769 9679559769 9680440869 9681331869 9682222869 9683113869 
/* We can also thread the whole mess, which for this case, is a 3.75 speed up
* (3 min to 48sec) with 8 cores (Intel 4/4).
*/
fcn palGT(n,N,sz){ palindromicGapfulW(n).drop(N-sz).walk(sz) } // worker thread
foreach N,sz in (T( T(100_000,1) )){
println("\nLast %d of %,d palindromic gapful numbers:".fmt(sz,N));
[1..9].apply('wrap(n){ palGT.future(n,N,sz) }) // create threads
.apply("noop") // wait for threads to finish
 : pgpp(_);
}
Output:
Last 1 of 100,000 palindromic gapful numbers:
1:   178788887871 
2:   278788887872 
3: 30878611687803 
4:  4833326233384 
5:   578789987875 
6: 60878611687806 
7: 74826144162847 
8: 80869688696808 
9: 96878077087869