Cyclops numbers
A cyclops number is a number with an odd number of digits that has a zero in the center, but nowhere else. They are named so in tribute to the one eyed giants Cyclops from Greek mythology.
Cyclops numbers can be found in any base. This task strictly is looking for cyclops numbers in base 10.
There are many different classifications of cyclops numbers with minor differences in characteristics. In an effort to head off a whole series of tasks with tiny variations, we will cover several variants here.
- Task
- Find and display here on this page the first 50 cyclops numbers in base 10 (all of the sub tasks are restricted to base 10).
- Find and display here on this page the first 50 prime cyclops numbers. (cyclops numbers that are prime.)
- Find and display here on this page the first 50 blind prime cyclops numbers. (prime cyclops numbers that remain prime when "blinded"; the zero is removed from the center.)
- Find and display here on this page the first 50 palindromic prime cyclops numbers. (prime cyclops numbers that are palindromes.)
- Stretch
- Find and display the first cyclops number greater than ten million (10,000,000) and the index (place) in the series where it is found.
- Find and display the first prime cyclops number greater than ten million (10,000,000) and the index (place) in the series where it is found.
- Find and display the first blind prime cyclops number greater than ten million (10,000,000) and the index (place) in the series where it is found.
- Find and display the first palindromic prime cyclops number greater than ten million (10,000,000) and the index (place) in the series where it is found.
(Note: there are no cyclops numbers between ten million and one hundred million, they need to have an odd number of digits)
- See also
ALGOL 68
Generates the sequence.
<lang algol68>BEGIN # show cyclops numbers - numbers with a 0 in the middle and no other 0 digits #
INT max prime = 100 000;
# sieve the primes to max prime #
PR read "primes.incl.a68" PR
[]BOOL prime = PRIMESIEVE max prime;
# returns TRUE if c is prime, FALSE otherwise #
PROC have a prime = ( INT c )BOOL:
IF c < 2 THEN FALSE
ELIF c < max prime
THEN prime[ c ]
ELSE # the cyclops number is too large for the sieve, use trial division #
BOOL possibly prime := ODD c;
FOR d FROM 3 BY 2 WHILE d * d <= c AND possibly prime DO possibly prime := c MOD d /= 0 OD;
possibly prime
FI # have a prime # ;
# arrays of the cyclops numbers we must show #
[ 1 : 50 ]INT first cyclops; INT cyclops count := 0;
[ 1 : 50 ]INT first prime cyclops; INT prime cyclops count := 0;
[ 1 : 50 ]INT first palindromic prime cyclops; INT palindromic prime cyclops count := 0;
[ 1 : 50 ]INT first blind prime cyclops; INT blind prime cyclops count := 0;
# notes c is a cyclops number, palindromic indicates whether it is palindromic or not #
# bc should be c c with the middle 0 removed #
# if c is one of the first 50 of various classifications, #
# it is stored in the appropriate array #
PROC have cyclops = ( INT c, BOOL palindromic, INT bc )VOID:
BEGIN
cyclops count +:= 1;
IF cyclops count <= UPB first cyclops THEN first cyclops[ cyclops count ] := c FI;
IF prime cyclops count < UPB first prime cyclops
OR ( palindromic prime cyclops count < UPB first palindromic prime cyclops AND palindromic )
OR blind prime cyclops count < UPB first blind prime cyclops
THEN
IF have a prime( c ) THEN
# have a prime cyclops #
IF prime cyclops count < UPB first prime cyclops THEN
first prime cyclops[ prime cyclops count +:= 1 ] := c
FI;
IF palindromic prime cyclops count < UPB first palindromic prime cyclops AND palindromic THEN
first palindromic prime cyclops[ palindromic prime cyclops count +:= 1 ] := c
FI;
IF blind prime cyclops count < UPB first blind prime cyclops THEN
IF have a prime( bc ) THEN
first blind prime cyclops[ blind prime cyclops count +:= 1 ] := c
FI
FI
FI
FI
END # have cyclops # ;
# prints a cyclops sequence #
PROC print cyclops = ( []INT seq, STRING legend, INT elements per line )VOID:
BEGIN
print( ( "The first ", whole( ( UPB seq - LWB seq ) + 1, 0 ), " ", legend, ":", newline, " " ) );
FOR i FROM LWB seq TO UPB seq DO
print( ( " ", whole( seq[ i ], -7 ) ) );
IF i MOD elements per line = 0 THEN print( ( newline, " " ) ) FI
OD;
print( ( newline ) )
END # print cyclops # ;
# generate the cyclops numbers #
# 0 is the first and only cyclops number with less than three digits #
have cyclops( 0, TRUE, 0 );
# generate the 3 digit cyclops numbers #
FOR f TO 9 DO
FOR b TO 9 DO
have cyclops( ( f * 100 ) + b
, f = b
, ( f * 10 ) + b
)
OD
OD;
# generate the 5 digit cyclops numbers #
FOR d1 TO 9 DO
FOR d2 TO 9 DO
INT d1200 = ( ( d1 * 10 ) + d2 ) * 100;
INT d12000 = d1200 * 10;
FOR d4 TO 9 DO
INT d40 = d4 * 10;
FOR d5 TO 9 DO
INT d45 = d40 + d5;
have cyclops( d12000 + d45
, d1 = d5 AND d2 = d4
, d1200 + d45
)
OD
OD
OD
OD;
# generate the 7 digit cyclops numbers #
FOR d1 TO 9 DO
FOR d2 TO 9 DO
FOR d3 TO 9 DO
INT d123000 = ( ( ( ( d1 * 10 ) + d2 ) * 10 ) + d3 ) * 1000;
INT d1230000 = d123000 * 10;
FOR d5 TO 9 DO
INT d500 = d5 * 100;
FOR d6 TO 9 DO
INT d560 = d500 + ( d6 * 10 );
FOR d7 TO 9 DO
INT d567 = d560 + d7;
have cyclops( d1230000 + d567
, d1 = d7 AND d2 = d6 AND d3 = d5
, d123000 + d567
)
OD
OD
OD
OD
OD
OD;
# show parts of the sequence #
print cyclops( first cyclops, "cyclops numbers", 10 );
print cyclops( first prime cyclops, "prime cyclops numbers", 10 );
print cyclops( first blind prime cyclops, "blind prime cyclops numbers", 10 );
print cyclops( first palindromic prime cyclops, "palindromic prime cyclops numbers", 10 )
END</lang>
- Output:
The first 50 cyclops numbers:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
The first 50 prime cyclops numbers:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
The first 50 blind prime cyclops numbers:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
The first 50 palindromic prime cyclops numbers:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
AWK
<lang AWK>
- syntax: GAWK -f CYCLOPS_NUMBERS.AWK
BEGIN {
n = 0
limit = 50
while (A134808_cnt < limit || A134809_cnt < limit || A329737_cnt < limit || A136098_cnt < limit) {
leng = length(n)
if (leng ~ /[13579]$/) {
middle_col = int(leng/2)+1
if (substr(n,middle_col,1) == 0 && gsub(/0/,"&",n) == 1) {
A134808_arr[++A134808_cnt] = n
if (is_prime(n)) {
A134809_arr[++A134809_cnt] = n
tmp = n
sub(/0/,"",tmp)
if (is_prime(tmp)) {
A329737_arr[++A329737_cnt] = n
}
if (reverse(n) == n) {
A136098_arr[++A136098_cnt] = n
}
}
}
}
n++
}
printf("Range: 0-%d\n\n",n-1)
show_array(A134808_arr,A134808_cnt,"A134808: Cyclops numbers")
show_array(A134809_arr,A134809_cnt,"A134809: Cyclops primes")
show_array(A329737_arr,A329737_cnt,"A329737: Cyclops primes that remain prime after being 'blinded'")
show_array(A136098_arr,A136098_cnt,"A136098: Prime palindromic cyclops numbers")
exit(0)
} function is_prime(x, i) {
if (x <= 1) {
return(0)
}
for (i=2; i<=int(sqrt(x)); i++) {
if (x % i == 0) {
return(0)
}
}
return(1)
} function reverse(str, i,rts) {
for (i=length(str); i>=1; i--) {
rts = rts substr(str,i,1)
}
return(rts)
} function show_array(arr,cnt,desc, count,i) {
printf("%s Found %d numbers within range\n",desc,cnt)
for (i=1; i<=limit; i++) {
printf("%7d%1s",arr[i],++count%10?"":"\n")
}
printf("\n")
} </lang>
- Output:
Range: 0-7820287
A134808: Cyclops numbers Found 407744 numbers within range
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
A134809: Cyclops primes Found 30270 numbers within range
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
A329737: Cyclops primes that remain prime after being 'blinded' Found 8892 numbers within range
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
A136098: Prime palindromic cyclops numbers Found 50 numbers within range
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
F#
This task uses Extensible Prime Generator (F#)
Cyclops function
<lang fsharp> // Cyclop numbers. Nigel Galloway: June 25th., 2021 let rec fG n g=seq{yield! g|>Seq.collect(fun i->g|>Seq.map(fun g->n*i+g)); yield! fG(n*10)(fN g)} let cyclops=seq{yield 0; yield! fG 100 [1..9]} let primeCyclops,blindCyclops=cyclops|>Seq.filter isPrime,Seq.zip(fG 100 [1..9])(fG 10 [1..9])|>Seq.filter(fun(n,g)->isPrime n && isPrime g)|>Seq.map fst let palindromicCyclops=let fN g=let rec fN g=[yield g%10; if g>9 then yield! fN(g/10)] in let n=fN g in n=List.rev n in primeCyclops|>Seq.filter fN </lang>
The tasks
- First 50 Cyclop numbers
<lang fsharp> cyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn "" </lang>
- Output:
0 101 102 103 104 105 106 107 108 109 201 202 203 204 205 206 207 208 209 301 302 303 304 305 306 307 308 309 401 402 403 404 405 406 407 408 409 501 502 503 504 505 506 507 508 509 601 602 603 604
- First 50 prime Cyclop numbers
<lang fsharp> primeCyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn "" </lang>
- Output:
101 103 107 109 307 401 409 503 509 601 607 701 709 809 907 11027 11047 11057 11059 11069 11071 11083 11087 11093 12011 12037 12041 12043 12049 12071 12073 12097 13033 13037 13043 13049 13063 13093 13099 14011 14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
- First 50 blind Cyclop numbers
<lang fsharp> blindCyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn "" </lang>
- Output:
101 103 107 109 307 401 503 509 601 607 701 709 809 907 11071 11087 11093 12037 12049 12097 13099 14029 14033 14051 14071 14081 14083 14087 15031 15053 15083 16057 16063 16067 16069 16097 17021 17033 17041 17047 17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
- First 50 palindromic prime Cyclop numbers
<lang fsharp> palindromicCyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn "" </lang>
- Output:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211 1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251 1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391 1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763 3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
- First Cyclop number > 10,000,000
<lang fsharp> let n=cyclops|>Seq.findIndex(fun n->n>10000000) in printfn "First Cyclop number > 10,000,000 is %d at index %d" (Seq.item n (cyclops)) n </lang>
- Output:
First Cyclop number > 10,000,000 is 111101111 at index 538084
- First prime Cyclop number > 10,000,000
<lang fsharp> let n=primeCyclops|>Seq.findIndex(fun n->n>10000000) in printfn "First prime Cyclop number > 10,000,000 is %d at index %d" (Seq.item n (primeCyclops)) n </lang>
- Output:
First prime Cyclop number > 10,000,000 is 111101129 at index 39319
- First blind Cyclop number > 10,000,000
<lang fsharp> let n=blindCyclops|>Seq.findIndex(fun n->n>10000000) in printfn "First blind Cyclop number > 10,000,000 is %d at index %d" (Seq.item n (blindCyclops)) n </lang>
- Output:
First blind Cyclop number > 10,000,000 is 111101161 at index 11393
- First palindromic prime Cyclop number > 10,000,000
<lang fsharp> let n=palindromicCyclops|>Seq.findIndex(fun n->n>10000000) in printfn "First palindromic prime Cyclop number > 10,000,000 is %d at index %d" (Seq.item n (palindromicCyclops)) n </lang>
- Output:
First palindromic prime Cyclop number > 10,000,000 is 114808411 at index 66
Factor
An attempt to make it fast has resulted in a lot of code, but here's the basic idea. Cyclops numbers are stored in four parts: an integer representing whatever is to the left of the 0, a right integer, the number of digits in either side, and the maximum number with that many digits. For example, T{ cyclops f 33 34 2 99 } represents the number 33034. The left and right are zeroless; that is, they are not allowed to have any zeros.
In order to find the successor to a given cyclops number:
- Increment the right integer to the next zeroless integer using the following formula (from OEIS):
a(n+1) = f(a(n)) with f(x) = 1 + if x mod 10 < 9 then x else 10*f([x/10])
- Unless right = max, in which case increment the left integer instead in the same manner, and reset the right integer back to
max / 9. - Unless left = max, in which case we move up to the next order of magnitude by adding
max/9 + 1to both sides, adding1to the number of digits, and setting the new max asmax := max * 10 + 9
Cyclops numbers in this form can be converted to integers (as late as possible) with int(c) = 10^(c.n + 1) * c.left + c.right
where n is the number of digits in either side.
Cyclops numbers can be converted to blind form with the same formula without the + 1.
<lang factor>USING: accessors formatting grouping io kernel lists lists.lazy math math.functions math.primes prettyprint sequences tools.memory.private tools.time ;
! ==========={[ Cyclops data type and operations ]}=============
TUPLE: cyclops left right n max ;
- <cyclops> ( -- cyclops ) 1 1 1 9 cyclops boa ;
- >cyclops< ( cyclops -- right left n )
[ right>> ] [ left>> ] [ n>> ] tri ;
M: cyclops >integer >cyclops< 1 + 10^ * + ;
- >blind ( cyclops -- n ) >cyclops< 10^ * + ;
- next-zeroless ( 9199 -- 9211 )
dup 10 mod 9 < [ 10 /i next-zeroless 10 * ] unless 1 + ;
- right++ ( cyclops -- cyclops' )
[ next-zeroless ] change-right ; inline
- left++ ( cyclops -- cyclops' )
[ next-zeroless ] change-left [ 9 /i ] change-right ;
- n++ ( cyclops -- cyclops' )
[ 1 + ] change-n [ 10 * 9 + ] change-max ;
- change-both ( cyclops quot -- cyclops' )
[ change-left ] keep change-right ; inline
- expand ( cyclops -- cyclops' )
dup max>> 9 /i 1 + '[ _ + ] change-both n++ ;
- carry ( cyclops -- cyclops' )
dup [ left>> ] [ max>> ] bi < [ left++ ] [ expand ] if ;
- succ ( cyclops -- next-cyclops )
dup [ right>> ] [ max>> ] bi < [ right++ ] [ carry ] if ;
! ============{[ List definitions & operations ]}===============
- lcyclops ( -- list ) <cyclops> [ succ ] lfrom-by ;
- lcyclops-int ( -- list ) lcyclops [ >integer ] lmap-lazy ;
- lprime-cyclops ( -- list )
lcyclops-int [ prime? ] lfilter ;
- lblind-prime-cyclops ( -- list )
lcyclops [ >integer prime? ] lfilter [ >blind prime? ] lfilter ;
- reverse-digits ( 123 -- 321 )
0 swap [ 10 /mod rot 10 * + swap ] until-zero ;
- lpalindromic-prime-cyclops ( -- list )
lcyclops [ [ left>> ] [ right>> ] bi reverse-digits = ] lfilter [ >integer prime? ] lfilter ;
- first>1e7 ( list -- elt index )
0 lfrom lzip [ first >integer 10,000,000 > ] lfilter car first2 [ >integer ] dip [ commas ] bi@ ;
! ====================={[ OUTPUT ]}=============================
- first50 ( list -- )
50 swap ltake [ >integer ] lmap list>array 10 group simple-table. ;
- show ( desc list -- )
desc desc "First 50 %s numbers:\n" printf
list [ first50 nl ] [
first>1e7
"First %s number > 10,000,000: %s - at (zero based) index: %s.\n\n\n" printf
] bi ;
"cyclops" lcyclops-int show "prime cyclops" lprime-cyclops show "blind prime cyclops" lblind-prime-cyclops show "palindromic prime cyclops" lpalindromic-prime-cyclops show
! Extra stretch? "One billionth cyclops number:" print 999,999,999 lcyclops lnth >integer commas print</lang>
- Output:
First 50 cyclops numbers: 101 102 103 104 105 106 107 108 109 201 202 203 204 205 206 207 208 209 301 302 303 304 305 306 307 308 309 401 402 403 404 405 406 407 408 409 501 502 503 504 505 506 507 508 509 601 602 603 604 605 First cyclops number > 10,000,000: 111,101,111 - at (zero based) index: 538,083. First 50 prime cyclops numbers: 101 103 107 109 307 401 409 503 509 601 607 701 709 809 907 11027 11047 11057 11059 11069 11071 11083 11087 11093 12011 12037 12041 12043 12049 12071 12073 12097 13033 13037 13043 13049 13063 13093 13099 14011 14029 14033 14051 14057 14071 14081 14083 14087 15013 15017 First prime cyclops number > 10,000,000: 111,101,129 - at (zero based) index: 39,319. First 50 blind prime cyclops numbers: 101 103 107 109 307 401 503 509 601 607 701 709 809 907 11071 11087 11093 12037 12049 12097 13099 14029 14033 14051 14071 14081 14083 14087 15031 15053 15083 16057 16063 16067 16069 16097 17021 17033 17041 17047 17053 17077 18047 18061 18077 18089 19013 19031 19051 19073 First blind prime cyclops number > 10,000,000: 111,101,161 - at (zero based) index: 11,393. First 50 palindromic prime cyclops numbers: 101 16061 31013 35053 38083 73037 74047 91019 94049 1120211 1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251 1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391 1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763 3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287 First palindromic prime cyclops number > 10,000,000: 114,808,411 - at (zero based) index: 66. One billionth cyclops number: 35,296,098,111
Go
<lang go>package main
import (
"fmt" "rcu" "strconv" "strings"
)
func findFirst(list []int) (int, int) {
for i, n := range list {
if n > 1e7 {
return n, i
}
}
return -1, -1
}
func reverse(s string) string {
chars := []rune(s)
for i, j := 0, len(chars)-1; i < j; i, j = i+1, j-1 {
chars[i], chars[j] = chars[j], chars[i]
}
return string(chars)
}
func main() {
ranges := [][2]int{
{0, 0}, {101, 909}, {11011, 99099}, {1110111, 9990999}, {111101111, 119101111},
}
var cyclops []int
for _, r := range ranges {
numDigits := len(fmt.Sprint(r[0]))
center := numDigits / 2
for i := r[0]; i <= r[1]; i++ {
digits := rcu.Digits(i, 10)
if digits[center] == 0 {
count := 0
for _, d := range digits {
if d == 0 {
count++
}
}
if count == 1 {
cyclops = append(cyclops, i)
}
}
}
}
fmt.Println("The first 50 cyclops numbers are:")
for i, n := range cyclops[0:50] {
fmt.Printf("%6s ", rcu.Commatize(n))
if (i+1)%10 == 0 {
fmt.Println()
}
}
n, i := findFirst(cyclops)
ns, is := rcu.Commatize(n), rcu.Commatize(i)
fmt.Printf("\nFirst such number > 10 million is %s at zero-based index %s\n", ns, is)
var primes []int
for _, n := range cyclops {
if rcu.IsPrime(n) {
primes = append(primes, n)
}
}
fmt.Println("\n\nThe first 50 prime cyclops numbers are:")
for i, n := range primes[0:50] {
fmt.Printf("%6s ", rcu.Commatize(n))
if (i+1)%10 == 0 {
fmt.Println()
}
}
n, i = findFirst(primes)
ns, is = rcu.Commatize(n), rcu.Commatize(i)
fmt.Printf("\nFirst such number > 10 million is %s at zero-based index %s\n", ns, is)
var bpcyclops []int
var ppcyclops []int
for _, p := range primes {
ps := fmt.Sprint(p)
split := strings.Split(ps, "0")
noMiddle, _ := strconv.Atoi(split[0] + split[1])
if rcu.IsPrime(noMiddle) {
bpcyclops = append(bpcyclops, p)
}
if ps == reverse(ps) {
ppcyclops = append(ppcyclops, p)
}
}
fmt.Println("\n\nThe first 50 blind prime cyclops numbers are:")
for i, n := range bpcyclops[0:50] {
fmt.Printf("%6s ", rcu.Commatize(n))
if (i+1)%10 == 0 {
fmt.Println()
}
}
n, i = findFirst(bpcyclops)
ns, is = rcu.Commatize(n), rcu.Commatize(i)
fmt.Printf("\nFirst such number > 10 million is %s at zero-based index %s\n", ns, is)
fmt.Println("\n\nThe first 50 palindromic prime cyclops numbers are:\n")
for i, n := range ppcyclops[0:50] {
fmt.Printf("%9s ", rcu.Commatize(n))
if (i+1)%8 == 0 {
fmt.Println()
}
}
n, i = findFirst(ppcyclops)
ns, is = rcu.Commatize(n), rcu.Commatize(i)
fmt.Printf("\n\nFirst such number > 10 million is %s at zero-based index %s\n", ns, is)
}</lang>
- Output:
The first 50 cyclops numbers are:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
First such number > 10 million is 111,101,111 at zero-based index 538,084
The first 50 prime cyclops numbers are:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11,027 11,047 11,057 11,059 11,069
11,071 11,083 11,087 11,093 12,011 12,037 12,041 12,043 12,049 12,071
12,073 12,097 13,033 13,037 13,043 13,049 13,063 13,093 13,099 14,011
14,029 14,033 14,051 14,057 14,071 14,081 14,083 14,087 15,013 15,017
First such number > 10 million is 111,101,129 at zero-based index 39,319
The first 50 blind prime cyclops numbers are:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11,071 11,087 11,093 12,037 12,049 12,097
13,099 14,029 14,033 14,051 14,071 14,081 14,083 14,087 15,031 15,053
15,083 16,057 16,063 16,067 16,069 16,097 17,021 17,033 17,041 17,047
17,053 17,077 18,047 18,061 18,077 18,089 19,013 19,031 19,051 19,073
First such number > 10 million is 111,101,161 at zero-based index 11,393
The first 50 palindromic prime cyclops numbers are:
101 16,061 31,013 35,053 38,083 73,037 74,047 91,019
94,049 1,120,211 1,150,511 1,160,611 1,180,811 1,190,911 1,250,521 1,280,821
1,360,631 1,390,931 1,490,941 1,520,251 1,550,551 1,580,851 1,630,361 1,640,461
1,660,661 1,670,761 1,730,371 1,820,281 1,880,881 1,930,391 1,970,791 3,140,413
3,160,613 3,260,623 3,310,133 3,380,833 3,460,643 3,470,743 3,590,953 3,670,763
3,680,863 3,970,793 7,190,917 7,250,527 7,310,137 7,540,457 7,630,367 7,690,967
7,750,577 7,820,287
First such number > 10 million is 114,808,411 at zero-based index 66
Haskell
<lang haskell>import Control.Monad (replicateM) import Data.Numbers.Primes (isPrime)
CYCLOPS NUMBERS --------------------
cyclops :: [Integer] cyclops = [0 ..] >>= flankingDigits
where
flankingDigits 0 = [0]
flankingDigits n =
fmap
(\s -> read s :: Integer)
( (fmap ((<>) . (<> "0")) >>= (<*>))
(replicateM n ['1' .. '9'])
)
blindPrime :: Integer -> Bool
blindPrime n =
let s = show n
m = quot (length s) 2
in isPrime $
(\s -> read s :: Integer)
(take m s <> drop (succ m) s)
palindromic :: Integer -> Bool
palindromic = ((==) =<< reverse) . show
TESTS -------------------------
main :: IO () main =
(putStrLn . unlines)
[ "First 50 Cyclops numbers – A134808:",
unwords (show <$> take 50 cyclops),
"",
"First 50 Cyclops primes – A134809:",
unwords $ take 50 [show n | n <- cyclops, isPrime n],
"",
"First 50 blind prime Cyclops numbers – A329737:",
unwords $
take
50
[show n | n <- cyclops, isPrime n, blindPrime n],
"",
"First 50 prime palindromic cyclops numbers – A136098:",
unwords $
take
50
[show n | n <- cyclops, isPrime n, palindromic n]
]</lang>
First 50 Cyclops numbers – A134808: 0 101 102 103 104 105 106 107 108 109 201 202 203 204 205 206 207 208 209 301 302 303 304 305 306 307 308 309 401 402 403 404 405 406 407 408 409 501 502 503 504 505 506 507 508 509 601 602 603 604 First 50 Cyclops primes – A134809: 101 103 107 109 307 401 409 503 509 601 607 701 709 809 907 11027 11047 11057 11059 11069 11071 11083 11087 11093 12011 12037 12041 12043 12049 12071 12073 12097 13033 13037 13043 13049 13063 13093 13099 14011 14029 14033 14051 14057 14071 14081 14083 14087 15013 15017 First 50 blind prime Cyclops numbers – A329737: 101 103 107 109 307 401 503 509 601 607 701 709 809 907 11071 11087 11093 12037 12049 12097 13099 14029 14033 14051 14071 14081 14083 14087 15031 15053 15083 16057 16063 16067 16069 16097 17021 17033 17041 17047 17053 17077 18047 18061 18077 18089 19013 19031 19051 19073 First 50 prime palindromic cyclops numbers – A136098: 101 16061 31013 35053 38083 73037 74047 91019 94049 1120211 1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251 1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391 1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763 3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
jq
Note that the indices shown in the output are 0-based, as jq's index origin is 0. <lang jq>
- Generic helper functions
def count(s): reduce s as $x (0; .+1);
- counting from 0
def enumerate(s): foreach s as $x (-1; .+1; [., $x]);
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
- Prime numbers
def is_prime:
if . == 2 then true
else
2 < . and . % 2 == 1 and
(. as $in
| (($in + 1) | sqrt) as $m
| [false, 3] | until( .[0] or .[1] > $m; [$in % .[1] == 0, .[1] + 2])
| .[0]
| not)
end ;
- Cyclops numbers
def iscyclops:
(tostring | explode) as $d | ($d|length) as $l | (($l + 1) / 2 - 1) as $m | ($l % 2 == 1) and $d[$m] == 48 and count($d[] | select(.== 48)) == 1 # "0"
- Generate a stream of cyclops numbers, in increasing numeric order, from 0
def cyclops:
# generate a stream of cyclops numbers with $n digits on each side of the central 0 def w: if . == 0 then "" else (.-1)|w as $left | $left + (range(1;10)|tostring) end; def c: w as $left | $left + "0" + w; range(0; infinite) | c | tonumber;
- Generate a stream of palindromic cyclops numbers, in increasing numeric order, from 0
def palindromiccyclops:
def r: explode|reverse|implode;
def c: . as $n
| if $n == 0 then "0"
elif $n == 1
then (range(1;10)|tostring) as $base
| $base + "0" + ($base | r)
else (range(pow(10;$n-1); pow(10; $n))|tostring|select(test("0")|not)) as $base
| $base + "0" + ($base | r)
end;
range(0; infinite) | c | tonumber;
- check that a cyclops number minus the 0 is prime
def cyclops_isblind:
(tostring | explode) as $d | ($d|length) as $l | (($l + 1) / 2 - 1) as $m | ((( $d[:$m] + $d[$m+1:] ) | implode | tonumber) | is_prime);
- check that a cyclops number is a palindrome
def cyclops_ispalindromic:
. as $in | (tostring | explode) as $d | ($d|length) as $l | (($l + 1) / 2 - 1) as $m | $d[:$m] == ( $d[$m+1:] | reverse) ;
</lang> The tasks <lang jq> def print5x10($width):
. as $a
| range(0;5) as $i
| reduce range(0;10) as $j (""; . + ($a[10*$i + $j] | lpad($width)));
def main_task($n):
"The first \($n) cyclops numbers are:",
([limit($n; cyclops)] | print5x10(7)),
"\nThe first \($n) prime cyclops numbers are:",
([limit($n; cyclops | select(is_prime))] | print5x10(7)),
"\nThe first \($n) blind prime cyclops numbers are:",
([limit($n; cyclops | select(is_prime and cyclops_isblind))] | print5x10(7)),
"\nThe first \($n) palindromic prime cyclops numbers are:",
([limit($n; cyclops | select(cyclops_ispalindromic and is_prime))] | print5x10(8)) ;
def stretch_task($big):
def pp: " \(.[1]) at index \(.[0]).";
"\nFirst cyclops greater than \($big):" +
(first(enumerate(cyclops) | select(.[1] > $big))|pp),
"\nThe next prime cyclops number after \($big):" +
(first(enumerate(cyclops | select(is_prime)) | select(.[1] | (. > $big)) ) | pp),
"\nThe first blind prime cyclops number greater than \($big):" +
(first(enumerate(cyclops | select(is_prime and cyclops_isblind)) | select(.[1] > $big) )|pp),
"\nThe first palindromic prime cyclops number greater than \($big):" +
(first(enumerate(palindromiccyclops | select(is_prime)) | select(.[1] > $big) ) | pp) ;
main_task(50), stretch_task(pow(10;7)) </lang>
- Output:
The first 50 cyclops numbers are:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
The first 50 prime cyclops numbers are:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
The first 50 blind prime cyclops numbers are:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
The first 50 palindromic prime cyclops numbers are:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
First cyclops greater than 10000000: 111101111 at index 538084.
The next prime cyclops number after 10000000: 111101129 at index 39319.
The first blind prime cyclops number greater than 10000000: 111101161 at index 11393.
The first palindromic prime cyclops number greater than 10000000: 114808411 at index 66.
Julia
Julia's indexes are 1 based, hence one greater than those of 0 based indices. <lang julia> print5x10(a, w = 8) = for i in 0:4, j in 1:10 print(lpad(a[10i + j], w), j == 10 ? "\n" : "") end
function iscyclops(n)
d = digits(n) l = length(d) return isodd(l) && d[l ÷ 2 + 1] == 0 && count(x -> x == 0, d) == 1
end
function isblindprimecyclops(n)
d = digits(n) l = length(d) m = l ÷ 2 + 1 (n == 0 || iseven(l) || d[m] != 0 || count(x -> x == 0, d) != 1) && return false return isprime(evalpoly(10, [d[1:m-1]; d[m+1:end]]))
end
function ispalindromicprimecyclops(n)
d = digits(n) l = length(d) return n > 0 && isodd(l) && d[l ÷ 2 + 1] == 0 && count(x -> x == 0, d) == 1 && d == reverse(d)
end
function findcyclops(N, iscycs, nextcandidate)
i, list = nextcandidate(-1), Int[]
while length(list) < N
iscycs(i) && push!(list, i)
i = nextcandidate(i)
end
return list
end
function nthcyclopsfirstafter(lowerlimit, iscycs, nextcandidate)
i, found = 0, 0
while true
if iscycs(i)
found += 1
i >= lowerlimit && break
end
i = nextcandidate(i)
end
return i, found
end
function testcyclops()
println("The first 50 cyclops numbers are:")
print5x10(findcyclops(50, iscyclops, x -> x + 1))
n, c = nthcyclopsfirstafter(10000000, iscyclops, x -> x + 1)
println("\nThe next cyclops number after 10,000,000 is $n at position $c.")
println("\nThe first 50 prime cyclops numbers are:")
print5x10(findcyclops(50, iscyclops, x -> nextprime(x + 1)))
n, c = nthcyclopsfirstafter(10000000, iscyclops, x -> nextprime(x + 1))
println("\nThe next prime cyclops number after 10,000,000 is $n at position $c.")
println("\nThe first 50 blind prime cyclops numbers are:")
print5x10(findcyclops(50, isblindprimecyclops, x -> nextprime(x + 1)))
n, c = nthcyclopsfirstafter(10000000, isblindprimecyclops, x -> nextprime(x + 1))
println("\nThe next prime cyclops number after 10,000,000 is $n at position $c.")
println("\nThe first 50 palindromic prime cyclops numbers are:")
print5x10(findcyclops(50, ispalindromicprimecyclops, x -> nextprime(x + 1)))
n, c = nthcyclopsfirstafter(10000000, ispalindromicprimecyclops, x -> nextprime(x + 1))
println("\nThe next prime cyclops number after 10,000,000 is $n at position $c.")
end
testcyclops()
</lang>
- Output:
The first 50 cyclops numbers are:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
The next cyclops number after 10,000,000 is 111101111 at position 538085.
The first 50 prime cyclops numbers are:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
The next prime cyclops number after 10,000,000 is 111101129 at position 39321.
The first 50 blind prime cyclops numbers are:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
The next prime cyclops number after 10,000,000 is 111101161 at position 11394.
The first 50 palindromic prime cyclops numbers are:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
The next prime cyclops number after 10,000,000 is 114808411 at position 67.
Mathematica / Wolfram Language
<lang Mathematica>a = Flatten[Table[FromDigits[{i, 0, j}], {i, 9}, {j, 9}], 1]; b = Flatten@Table[FromDigits@{i, j}, {i, 9}, {j, 9}]; c = Flatten@Table[FromDigits@{i, j, k}, {i, 9}, {j, 9}, {k, 9}];
blindPrimeQ[n_] :=
Block[{digits = IntegerDigits@n, m},
m = Floor[Length[digits]/2];
PrimeQ[FromDigits@Join[digits;; m, digits-m ;;]]]
palindromeQ[n_] :=
Block[{digits = IntegerDigits@n}, digits === Reverse[digits]]
cyclopsQ[n_] :=
Block[{digits = IntegerDigits@n, len, ctr},
len = Length[digits];
ctr = Ceiling[len/2];
And @@ {Mod[len, 2] == 1, ctr > 0, digitsctr == 0,
FreeQ[Drop[digits, {ctr}], 0]}]
cyclops = (* all Cyclops numbers with 3, 5 or 7 digits *)
Flatten@{a,
Outer[
FromDigits@Flatten@{IntegerDigits@#1, 0, IntegerDigits@#2} &, b,
b],
Outer[
FromDigits@Flatten@{IntegerDigits@#1, 0, IntegerDigits@#2} &, c,
c]};
x = NestWhile[NextPrime, 10^8, ! (cyclopsQ@# && PrimeQ@#) &];
Labeled[Partition[Flatten[{0, a}];; 50, 10] //
TableForm, "First 50 Cyclop numbers", Top]
Labeled[Partition[(primeCyclops = Cases[cyclops, _?PrimeQ]);; 50,
10] // TableForm, "First 50 prime cyclops numbers", Top]
Labeled[Partition[(blind = Cases[primeCyclops, _?blindPrimeQ])[[;;
50]], 10] // TableForm, "First 50 blind prime cyclops numbers", Top]
Labeled[Partition[(p = Cases[primeCyclops, _?palindromeQ]);; 50,
10] // TableForm, "First 50 palindromic prime Cyclops Numbers", Top]
Labeled[TableForm[{{x,
Length@primeCyclops}, {NestWhile[NextPrime,
x + 1, ! (cyclopsQ@# && PrimeQ@# && blindPrimeQ@#) &],
Length@blind}, {NestWhile[NextPrime,
x + 1, ! (cyclopsQ@# && PrimeQ@# && palindromeQ@#) &],
Length@p}},
TableHeadings -> {{"Prime", "Blind Prime",
"Palindromic Prime"}, {"Value",
"Index"}}], "First Cyclops numeber > 10,000,000", Top]</lang>
- Output:
First 50 Cyclop numbers 0 101 102 103 104 105 106 107 108 109 201 202 203 204 205 206 207 208 209 301 302 303 304 305 306 307 308 309 401 402 403 404 405 406 407 408 409 501 502 503 504 505 506 507 508 509 601 602 603 604
First 50 prime cyclops numbers 101 103 107 109 307 401 409 503 509 601 607 701 709 809 907 11027 11047 11057 11059 11069 11071 11083 11087 11093 12011 12037 12041 12043 12049 12071 12073 12097 13033 13037 13043 13049 13063 13093 13099 14011 14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
First 50 blind prime cyclops numbers 101 103 107 109 307 401 503 509 601 607 701 709 809 907 11071 11087 11093 12037 12049 12097 13099 14029 14033 14051 14071 14081 14083 14087 15031 15053 15083 16057 16063 16067 16069 16097 17021 17033 17041 17047 17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
First 50 palindromic prime Cyclops Numbers 101 16061 31013 35053 38083 73037 74047 91019 94049 1120211 1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251 1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391 1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763 3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
First Cyclops numeber > 10,000,000
Value IndexPrime 111101129 39319 Blind Prime 111101161 11393 Palindromic Prime 114808411 66
Nim
<lang Nim>import strutils, times
const Ranges = [0..0, 101..909, 11011..99099, 1110111..9990999, 111101111..999909999]
func isCyclops(d: string): bool =
d[d.len shr 1] == '0' and d.count('0') == 1
func isPrime(n: Natural): bool =
if n < 2: return if n mod 2 == 0: return n == 2 if n mod 3 == 0: return n == 3 var d = 5 while d * d <= n: if n mod d == 0: return false inc d, 2 if n mod d == 0: return false inc d, 4 return true
func isBlind(d: string): bool =
var d = d let m = d.len shr 1 result = (d[0..m-1] & d[m+1..^1]).parseInt().isPrime
func isPalindromic(d: string): bool =
for i in 1..d.len: if d[i-1] != d[^i]: return result = true
iterator cyclops(): (int, int) =
var count = 0
for r in Ranges:
for n in r:
if ($n).isCyclops:
inc count
yield (count, n)
iterator primeCyclops(): (int, int) =
var count = 0
for (_, n) in cyclops():
if n.isPrime:
inc count
yield (count, n)
iterator blindPrimeCyclops(): (int, int) =
var count = 0
for (_, n) in primeCyclops():
if ($n).isBlind:
inc count
yield (count, n)
iterator palindromicPrimeCyclops(): (int, int) =
var count = 0
for r in Ranges:
for n in r:
let d = $n
if d.isCyclops and d.isPalindromic and n.isPrime:
inc count
yield (count, n)
let t0 = cpuTime()
echo "List of first 50 cyclops numbers:" for i, n in cyclops():
stdout.write ($n).align(3), if i mod 10 == 0: '\n' else: ' ' if i == 50: break
echo "\nList of first 50 prime cyclops numbers:" for i, n in primeCyclops():
stdout.write ($n).align(5), if i mod 10 == 0: '\n' else: ' ' if i == 50: break
echo "\nList of first 50 blind prime cyclops numbers:" for i, n in blindPrimeCyclops():
stdout.write ($n).align(5), if i mod 10 == 0: '\n' else: ' ' if i == 50: break
echo "\nList of first 50 palindromic prime cyclops numbers:" for i, n in palindromicPrimeCyclops():
stdout.write ($n).align(7), if i mod 10 == 0: '\n' else: ' ' if i == 50: break
for i, n in cyclops():
if n > 10_000_000:
echo "\nFirst cyclops number greater than ten million is ",
($n).insertSep(), " at 1-based index: ", i
break
for i, n in primeCyclops():
if n > 10_000_000:
echo "\nFirst prime cyclops number greater than ten million is ",
($n).insertSep(), " at 1-based index: ", i
break
for i, n in blindPrimeCyclops():
if n > 10_000_000:
echo "\nFirst blind prime cyclops number greater than ten million is ",
($n).insertSep(), " at 1-based index: ", i
break
for i, n in palindromicPrimeCyclops():
if n > 10_000_000:
echo "\nFirst palindromic prime cyclops number greater than ten million is ",
($n).insertSep(), " at 1-based index: ", i
break
echo "\nExecution time: ", (cpuTime() - t0).formatFloat(ffDecimal, precision = 3), " seconds."</lang>
- Output:
List of first 50 cyclops numbers:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
List of first 50 prime cyclops numbers:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
List of first 50 blind prime cyclops numbers:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
List of first 50 palindromic prime cyclops numbers:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
First cyclops number greater than ten million is 111_101_111 at 1-based index: 538085
First prime cyclops number greater than ten million is 111_101_129 at 1-based index: 39320
First blind prime cyclops number greater than ten million is 111_101_161 at 1-based index: 11394
First palindromic prime cyclops number greater than ten million is 114_808_411 at 1-based index: 67
Execution time: 2.936 seconds.
Pascal
Using left and reight part of cyclops number in base 9 added to every digit 1 [0..8]->[1..9]
Simple trial division for isprime is very slow.
Try to find one billionth like Cyclops_numbers#Factor
Added conversion of zero-based index to cyclops number
verified List of palindromic prime cyclops numbers
<lang pascal>program cyclops;
{$IFDEF FPC}
{$MODE DELPHI}
{$OPTIMIZATION ON,ALL}
{$CodeAlign proc=32,loop=1}
{$ENDIF} //extra https://oeis.org/A136098/b136098.txt take ~37 s( TIO.RUN ) uses
sysutils;
const
BIGLIMIT = 10*1000*1000;
type
//number in base 9
tnumdgts = array[0..10] of byte;
tpnumdgts = pByte;
tnum9 = packed record
nmdgts : tnumdgts;
nmMaxDgtIdx :byte;
nmNum : uint32;
end;
tCN = record
cnRight,
cnLeft : tNum9;
cnNum : Uint64;
cndigits,
cnIdx : Uint32;
end;
tCyclopsList = array of Uint64;
procedure InitCycNum(var cn:tCN);forward;
var
cnMin,cnPow10Shift,cnPow9 : array[0..15] of Uint64; Cyclops :tCyclopsList;
function IndexToCyclops(n:Uint64):tCN; //zero-based index var
dgtCnt,i,num : UInt32; q,p9: Uint64;
Begin
InitCycNum(result); if n = 0 then EXIT; result.cnIdx := n; dgtCnt := 0;
repeat
p9 := sqr(cnPow9[dgtCnt]);
if n < p9 then
break;
n -= p9;
inc(dgtCnt)
until dgtcnt>10;
dec(dgtCnt);
with result do
Begin
with cnRight do
Begin
nmMaxDgtIdx := dgtCnt;
For i := 0 to dgtCnt do
begin
q := n DIV 9;
nmdgts[i] := n-9*q;
n := q;
end;
num :=0;
For i := dgtcnt downto 0 do
num := num*10+nmdgts[i]+1;
nmNum:= num;
cnNum := num;
end;
with cnLeft do
Begin
nmMaxDgtIdx := dgtCnt;
For i := 0 to dgtCnt do
begin
q := n DIV 9;
nmdgts[i] := n-9*q;
n := q;
end;
num :=0;
For i := dgtcnt downto 0 do
num := num*10+nmdgts[i]+1;
nmNum:= num;
cnNum += num*cnPow10Shift[dgtCnt];
cndigits := dgtCnt;
end;
end;
end;
procedure Out_Cyclops(const cl:tCyclopsList;colw,colc:NativeInt); var
i,n : NativeInt;
Begin
n := High(cl);
If n > 100 then
n := 100;
For i := 0 to n do
begin
write(cl[i]:colw);
if (i+1) mod colc = 0 then
writeln;
end;
if (i+1) mod colc <> 0 then
writeln;
if n< High(cl) then
writeln(High(cl)+1,' : ',cl[High(cl)]);
end;
procedure InitCnMinPow; //min = (0,1,11,111,1111,11111,111111,1111111,11111111,...); var
i,min,pow,pow9 : Uint64;
begin
min := 0; pow := 100; pow9 := 1; For i :=0 to High(cnMin) do begin cnMin[i] := min; min := 10*min+1; cnPow10Shift[i] := pow; pow *= 10; cnPow9[i] := pow9; pow9 *= 9; end;
end;
procedure ClearNum9(var tn:tNum9;idx:Uint32); begin
fillchar(tn,SizeOf(tn),#0); tn.nmNum := cnMin[idx+1];
end;
Procedure InitCycNum(var cn:tCN); Begin
with cn do Begin cndigits := 0; ClearNum9(cnLeft,0); ClearNum9(cnRight,0); cnNum := 0; cnIdx := 0; end;
end;
procedure IncNum9(var tn:tNum9); var
idx,fac,n: Uint32;
begin
idx := 0;
with tn do
Begin
fac := 1;
n := nmdgts[0]+1;
inc(nmNum);
repeat
if n < 9 then
break;
inc(nmNum,fac);
nmdgts[idx] :=0;
inc(idx);
fac *= 10;
n := nmdgts[idx]+1;
until idx > nmMaxDgtIdx;
If idx > High(nmdgts) then
EXIT;
nmdgts[idx] := n;
if nmMaxDgtIdx<Idx then
nmMaxDgtIdx := Idx;
end;
end;
procedure NextCycNum(var cycnum:tCN); begin
with cycnum do
Begin
if cnIdx <> 0 then
begin
//increment right digits
IncNum9(cnRight);
if cnRight.nmMaxDgtIdx > cndigits then
Begin
//set right digits to minimum
ClearNum9(cnRight,cndigits);
//increment left digits
IncNum9(cnLeft);
//One more digit ?
if cnLeft.nmMaxDgtIdx > cndigits then
Begin
inc(cndigits);
ClearNum9(cnLeft,cndigits);
ClearNum9(cnRight,cndigits);
if cndigits>High(tnumdgts) then
cndigits := High(tnumdgts);
end;
end;
cnNum := cnLeft.nmNum*cnPow10Shift[cndigits]+cnRight.nmNUm;
inc(cnIdx);
end
else
Begin
cnNum := 101;
cnIdx := 1;
end;
end;
end;
procedure MakePalinCycNum(var cycnum:tCN); //make right to be palin of left var
n,dgt : Uint32; i,j:NativeInt;
Begin
n := 0;
with cycnum do
Begin
i := 0;
For j := cnDigits downto 0 do
begin
dgt := cnLeft.nmdgts[i];
cnRight.nmdgts[j] := dgt;
n := 10*n+(dgt+1);
inc(i);
end;
cnRight.nmNum := n;
cnNum := cnLeft.nmNum*cnPow10Shift[cndigits]+n;
end;
end;
procedure IncLeftCn(var cn:tCN); Begin
with cn do
Begin
//set right digits to minimum
ClearNum9(cnRight,cndigits);
//increment left digits
IncNum9(cnLeft);
//One more digit ?
if cnLeft.nmMaxDgtIdx > cndigits then
Begin
inc(cndigits);
ClearNum9(cnLeft,cndigits);
ClearNum9(cnRight,cndigits);
if cndigits>High(tnumdgts) then
cndigits := High(tnumdgts);
end;
cnNum := cnLeft.nmNum*cnPow10Shift[cndigits]+cnRight.nmNUm;
end;
end;
function isPalinCycNum(const cycnum:tCN):boolean; var
i,j:NativeInt;
Begin
with cycnum do
Begin
i := cnDigits;
j := 0;
repeat
result := (cnRight.nmdgts[i]=cnLeft.nmdgts[j]);
if not(result) then
BREAK;
dec(i);
inc(j);
until i<0;
end;
end;
function FirstCyclops(cnt:NativeInt):tCN; var
i: NativeInt;
begin
setlength(Cyclops,cnt); i := 0; InitCycNum(result); while i < cnt do begin Cyclops[i] := result.cnNum; inc(i); NextCycNum(result); end; repeat NextCycNum(result); inc(i); until result.cnNum> BIGLIMIT;
end;
function isPrime(n:Uint64):boolean; var
p,q : Uint64;
Begin {
if n< 4 then
Begin
if n < 2 then
EXIT(false);
EXIT(true);
end;
if n = 5 then
exit(true);}
if (n AND 1 = 0) then
EXIT(false);
q := n div 3;
if n - 3*q = 0 then
EXIT(false);
p := 5;
{$CodeAlign loop=1}
repeat
q := n div p;
if n-q*p = 0 then
EXIT(false);
p += 2;
q := n div p;
if n-q*p = 0 then
EXIT(false);
if q < p then
break;
p += 4;
until false;
EXIT(true);
end;
function FirstPrimeCyclops(cnt:NativeInt):tCN; var
i: NativeInt;
begin
i := 0;
setlength(Cyclops,cnt);
InitCycNum(result);
while i<cnt do
begin
if isPrime(result.cnNum) then
Begin
Cyclops[i] := result.cnNum;
inc(i);
end;
NextCycNum(result);
end;
repeat
if isPrime(result.cnNum) then
begin;
inc(i);
if result.cnNum > BIGLIMIT then
BREAK;
end;
NextCycNum(result);
until false;
result.cnIdx := i;
end;
function FirstBlindPrimeCyclops(cnt:NativeInt):tCN; var
n: Uint64; i: NativeInt; isPr:Boolean;
begin
i := 0;
setlength(Cyclops,cnt);
InitCycNum(result);
while i< cnt do
begin
with result do
if isPrime(cnNum) then
Begin
n:= cnRight.nmNum;
if cndigits > 0 then
n += cnLeft.nmNum*cnPow10Shift[cndigits-1]
else
n += cnLeft.nmNum*10;
if isPrime(n) then
Begin
Cyclops[i] := cnNum;
inc(i);
end;
end;
NextCycNum(result);
end;
repeat
with result do
if isPrime(cnNum) then
Begin
n:= cnRight.nmNum;
if cndigits > 0 then
n += cnLeft.nmNum*cnPow10Shift[cndigits-1]
else
n += cnLeft.nmNum*10;
isPr:= isPrime(n);
inc(i,Ord(isPr));
if isPr AND (cnNum > BIGLIMIT) then
BREAK;
end;
NextCycNum(result);
until false;
result.cnIdx := i;
end;
function FirstPalinPrimeCyclops(cnt:NativeInt):tCN; var
i: NativeInt;
begin
i := 0;
setlength(Cyclops,cnt);
InitCycNum(result);
while i< cnt do
Begin
MakePalinCycNum(result);
with result do
if isPrime(cnNum) then
Begin
Cyclops[i] := cnNum;
inc(i);
end;
IncLeftCn(result);
while Not(result.cnLeft.nmdgts[result.cnDigits]+1 in [1,3,7,9]) do
IncLeftCn(result);
end;
repeat
MakePalinCycNum(result);
with result do
if isPrime(cnNum) then
begin
inc(i);
if cnNum >BIGLIMIT then
break;
end;
IncLeftCn(result);
while Not(result.cnLeft.nmdgts[result.cnDigits]+1 in [1,3,7,9]) do
IncLeftCn(result);
until false;
result.cnIdx := i;
end;
var
cycnum:tCN; T0 : Int64; cnt : NativeUint;
begin
InitCnMinPow;
cnt := 50;
writeln('The first ',cnt,' cyclops numbers are:');
cycnum := FirstCyclops(cnt);
Out_Cyclops(Cyclops,5,10);
writeln('First such number > ',BIGLIMIT,' is ',cycnum.cnNum,
' at zero-based index ',cycnum.cnIdx);
writeln;
cnt := 50;
writeln('The first ',cnt,' prime cyclops numbers are:');
T0 := GetTickCount64;
cycnum := FirstPrimeCyclops(cnt);
T0 := GetTickCOunt64-T0;
Out_Cyclops(Cyclops,7,10);
writeln('First such number > ',BIGLIMIT,' is ',cycnum.cnNum,
' at one-based index ',cycnum.cnIdx);
writeln(cycnum.cnIdx,'.th = ',cycnum.cnNum,' in ',T0/1000:6:3,' s');
writeln;
cnt := 50;
writeln('The first ',cnt,' blind prime cyclops numbers are:');
T0 := GetTickCount64;
cycnum := FirstBlindPrimeCyclops(cnt);
T0 := GetTickCOunt64-T0;
Out_Cyclops(Cyclops,7,10);
writeln('First such number > ',BIGLIMIT,' is ',cycnum.cnNum,
' at one-based index ',cycnum.cnIdx);
writeln(cycnum.cnIdx,'.th = ',cycnum.cnNum,' in ',T0/1000:6:3,' s');
writeln;
cnt := 50;
writeln('The first ',cnt,' palindromatic prime cyclops numbers are:');
cycnum := FirstPalinPrimeCyclops(cnt);
Out_Cyclops(Cyclops,15,5);
writeln('First such number > ',BIGLIMIT,' is ',cycnum.cnNum,
' at one-based index ',cycnum.cnIdx);
writeln;
cnt := 100;
repeat
write(cnt:17,'.th = ');
if cnt <= 10*1000 then
Begin
InitCycNum(cycnum);
repeat
NextCycNum(cycnum);
until cycnum.cnIdx = cnt;
write(cycnum.cnNum);
end;
cycnum:= IndexToCyclops(cnt);
writeln(' calc ',cycnum.cnNum);
cnt *= 10;
until cnt >1000*1000*1000*1000*1000;
end. </lang>
- Output:
TIO.RUN
The first 50 cyclops numbers are:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
First such number > 10000000 is 111101111 at zero-based index 538084
The first 50 prime cyclops numbers are:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
First such number > 10000000 is 111101129 at one-based index 39320
39320.th = 111101129 in 0.311 s
The first 50 blind prime cyclops numbers are:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
First such number > 10000000 is 111101161 at one-based index 11394
11394.th = 111101161 in 0.339 s
The first 50 palindromatic prime cyclops numbers are:
101 16061 31013 35053 38083
73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521
1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661
1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133
3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137
7540457 7630367 7690967 7750577 7820287
First such number > 10000000 is 114808411 at one-based index 67
100.th = 11031 calc 11031
1000.th = 23041 calc 23041
10000.th = 1150651 calc 1150651
100000.th = calc 2630161
1000000.th = calc 118804671
10000000.th = calc 298302381
100000000.th = calc 12382046191
1000000000.th = calc 35296098111
10000000000.th = calc 1287360779121
100000000000.th = calc 4171140671231
1000000000000.th = calc 135781601473441
10000000000000.th = calc 484596705297751
100000000000000.th = calc 14431574037376261
1000000000000000.th = calc 57737327021238771
Real time: 0.794 s CPU share: 99.38 %
Perl
<lang perl>use strict; use warnings; use feature 'say'; use ntheory 'is_prime'; use List::AllUtils 'firstidx';
sub comma { reverse ((reverse shift) =~ s/(.{3})/$1,/gr) =~ s/^,//r }
my @cyclops = 0; for my $exp (0..3) {
my @oom = grep { ! /0/ } 10**$exp .. 10**($exp+1)-1;
for my $l (@oom) {
for my $r (@oom) {
push @cyclops, $l . '0' . $r;
}
}
}
my @prime_cyclops = grep { is_prime $_ } @cyclops; my @prime_blind = grep { is_prime $_ =~ s/0//r } @prime_cyclops; my @prime_palindr = grep { $_ eq reverse $_ } @prime_cyclops;
my $upto = 50; my $over = 10_000_000;
for (
[, @cyclops],
['prime', @prime_cyclops],
['blind prime', @prime_blind],
['palindromic prime', @prime_palindr]) {
my($text,@values) = @$_;
my $i = firstidx { $_ > $over } @values;
say "First $upto $text cyclops numbers:\n" .
(sprintf "@{['%8d' x $upto]}", @values[0..$upto-1]) =~ s/(.{80})/$1\n/gr;
printf "First $text number > %s: %s at (zero based) index: %s\n\n", map { comma($_) } $over, $values[$i], $i;
}</lang>
- Output:
First 50 cyclops numbers:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
First number > 10,000,000: 111,101,111 at (zero based) index: 538,084
First 50 prime cyclops numbers:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
First prime number > 10,000,000: 111,101,129 at (zero based) index: 39,319
First 50 blind prime cyclops numbers:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
First blind prime number > 10,000,000: 111,101,161 at (zero based) index: 11,393
First 50 palindromic prime cyclops numbers:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
First palindromic prime number > 10,000,000: 114,808,411 at (zero based) index: 66
Phix
You can run this online here (expect a blank screen for about 8s).
with javascript_semantics atom t0 = time() function bump(sequence half) -- add a digit to valid halves -- eg {0} --> {1..9} (no zeroes) -- --> {11..99} ("") -- --> {111..999}, etc sequence res = {} for i=1 to length(half) do integer hi = half[i]*10 for digit=1 to 9 do res &= hi+digit end for end for return res end function procedure cyclops(string s="") sequence res = {}, half = {0} -- valid digits, see bump() integer left = 1, -- half[] before the 0 right = 0, -- half[] after the 0 hlen = 1, -- length(half) cpow = 10, -- cyclops power (of 10) bcpow = 1, -- blind cyclops power cn = 0 -- cyclops number (scratch) bool valid = false, bPrime = match("prime",s), bBlind = match("blind",s), bPalin = match("palindromic",s) while length(res)<50 or cn<=1e7 or not valid do right += 1 if right>hlen then right = 1 left += 1 if left>hlen then half = bump(half) hlen = length(half) cpow *= 10 bcpow *= 10 left = 1 end if end if integer lh = half[left], rh = half[right] cn = lh*cpow+rh -- cyclops number valid = not bPrime or is_prime(cn) if valid and bBlind then valid = is_prime(lh*bcpow+rh) end if if valid and bPalin then valid = sprintf("%d",lh) == reverse(sprintf("%d",rh)) end if if valid then res = append(res,sprintf("%7d",cn)) end if end while printf(1,"First 50 %scyclops numbers:\n%s\n",{s,join_by(res[1..50],1,10)}) printf(1,"First %scyclops number > 10,000,000: %s at (one based) index: %d\n\n", {s,res[$],length(res)}) end procedure cyclops() cyclops("prime ") cyclops("blind prime ") cyclops("palindromic prime ") ?elapsed(time()-t0)
- Output:
First 50 cyclops numbers:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
First cyclops number > 10,000,000: 111101111 at (one based) index: 538085
First 50 prime cyclops numbers:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
First prime cyclops number > 10,000,000: 111101129 at (one based) index: 39320
First 50 blind prime cyclops numbers:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
First blind prime cyclops number > 10,000,000: 111101161 at (one based) index: 11394
First 50 palindromic prime cyclops numbers:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
First palindromic prime cyclops number > 10,000,000: 114808411 at (one based) index: 67
"7.6s"
billionth cyclops number
Finding the billionth number near-instantly. You can run this online here
with javascript_semantics atom t0 = time() function bump(sequence half) -- (exactly the same as above) -- add a digit to valid halves -- eg {0} --> {1..9} (no zeroes) -- --> {11..99} ("") -- --> {111..999}, etc sequence res = {} for i=1 to length(half) do integer hi = half[i]*10 for digit=1 to 9 do res &= hi+digit end for end for return res end function sequence half = {0} -- valid digits, see bump() integer hlen = 1, cpow = 10 -- cyclops power (of 10) integer n = 1_000_000_000 atom hlen2 = 1 while n>hlen2 do n -= hlen2 half = bump(half) hlen = length(half) hlen2 = power(hlen,2) cpow *= 10 end while integer left = floor(n/hlen)+1, -- half[] before the 0 right = remainder(n,hlen)+1 -- half[] after the 0 string fmt = "The 1,000,000,000th cyclops number is %d (%s)\n", e = elapsed(time()-t0) printf(1,fmt,{half[left]*cpow+half[right],e})
- Output:
The 1,000,000,000th cyclops number is 35296098111 (0s)
Raku
<lang perl6>use Lingua::EN::Numbers;
my @cyclops = 0, |flat lazy ^∞ .map: -> $exp {
my @oom = (exp($exp, 10) ..^ exp($exp + 1, 10)).grep: { !.contains: 0 }
|@oom.hyper.map: { $_ ~ 0 «~« @oom }
}
my @prime-cyclops = @cyclops.grep: { .is-prime };
for , @cyclops,
'prime ', @prime-cyclops,
'blind prime ', @prime-cyclops.grep( { .trans('0' => ).is-prime } ),
'palindromic prime ', @prime-cyclops.grep( { $_ eq .flip } )
-> $type, $iterator {
say "\n\nFirst 50 {$type}cyclops numbers:\n" ~ $iterator[^50].batch(10)».fmt("%7d").join("\n") ~
"\n\nFirst {$type}cyclops number > 10,000,000: " ~ comma($iterator.first: * > 1e7 ) ~
" - at (zero based) index: " ~ comma $iterator.first: * > 1e7, :k;
}</lang>
- Output:
First 50 cyclops numbers:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
First cyclops number > 10,000,000: 111,101,111 - at (zero based) index: 538,084
First 50 prime cyclops numbers:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
First prime cyclops number > 10,000,000: 111,101,129 - at (zero based) index: 39,319
First 50 blind prime cyclops numbers:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
First blind prime cyclops number > 10,000,000: 111,101,161 - at (zero based) index: 11,393
First 50 palindromic prime cyclops numbers:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
First palindromic prime cyclops number > 10,000,000: 114,808,411 - at (zero based) index: 66
REXX
<lang rexx>/*REXX pgm finds 1st N cyclops (Θ) #s, Θ primes, blind Θ primes, palindromic Θ primes*/ parse arg n cols . /*obtain optional argument from the CL.*/ if n== | n=="," then n= 50 /*Not specified? Then use the default.*/ if cols== | cols=="," then cols= 10 /* " " " " " " */ call genP /*build array of semaphores for primes.*/ w= max(10, length( commas(@.#) ) ) /*max width of a number in any column. */ pri?= 0; bli?= 0; pal?= 0; call 0 ' first ' commas(n) " cyclops numbers" pri?= 1; bli?= 0; pal?= 0; call 0 ' first ' commas(n) " prime cyclops numbers" pri?= 1; bli?= 1; pal?= 0; call 0 ' first ' commas(n) " blind prime cyclops numbers" pri?= 1; bli?= 0; pal?= 1; call 0 ' first ' commas(n) " palindromic prime cyclops numbers" exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? /*──────────────────────────────────────────────────────────────────────────────────────*/ 0: parse arg title; idx= 1 /*get the title of this output section.*/
say ' index │'center(title, 1 + cols*(w+1) ) /*display the output title. */
say '───────┼'center("" , 1 + cols*(w+1), '─') /* " " " separator*/
finds= 0; $= /*the number of finds (so far); $ list.*/
do j=0 until finds== n; L= length(j) /*find N cyclops numbers, start at 101.*/
if L//2==0 then do; j= left(1, L+1, 0) /*Is J an even # of digits? Yes, bump J*/
iterate /*use a new J that has odd # of digits.*/
end
z= pos(0, j); if z\==(L+1)%2 then iterate /* " " " " (zero in mid)? " */
if pos(0, j, z+1)>0 then iterate /* " " " " (has two 0's)? " */
if pri? then if \!.j then iterate /*Need a cyclops prime? Then skip.*/
if bli? then do; ?= space(translate(j, , 0), 0) /*Need a blind cyclops prime ?*/
if \!.? then iterate /*Not a blind cyclops prime? Then skip.*/
end
if pal? then do; r= reverse(j) /*Need a palindromic cyclops prime? */
if r\==j then iterate /*Cyclops number not palindromic? Skip.*/
if \!.r then iterate /* " palindrome not prime? " */
end
finds= finds + 1 /*bump the number of palindromic primes*/
$= $ right( commas(j), w) /*add a palindromic prime ──► $ list.*/
if finds//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/
if $\== then say center(idx, 7)"│" substr($, 2) /*possible show residual output.*/
say '───────┴'center("" , 1 + cols*(w+1), '─'); say
return
/*──────────────────────────────────────────────────────────────────────────────────────*/ genP: !.= 0; hip= 7890987 - 1 /*placeholders for primes (semaphores).*/
@.1=2; @.2=3; @.3=5; @.4=7; @.5=11; @.6=13 /*define some low primes. */
!.2=1; !.3=1; !.5=1; !.7=1; !.11=1; !.13=1 /* " " " " flags. */
#= 6; sq.#= @.# ** 2 /*number of primes so far; prime square*/
do j=@.#+2 by 2 for max(0, hip%2-@.#%2-1) /*find odd primes from here on. */
parse var j -1 _ /*get the last dec. digit of J.*/
if _==5 then iterate; if j// 3==0 then iterate /*÷ by 5? ÷ by 3? Skip.*/
if j// 7==0 then iterate; if j//11==0 then iterate /*÷ " 7? ÷ by 11? " */
do k=6 while sq.k<=j /* [↓] divide by the known odd primes.*/
if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */
end /*k*/ /* [↑] only process numbers ≤ √ J */
#= #+1; @.#= j; sq.#= j*j; !.j= 1 /*bump # Ps; assign next P; P sq; P# */
end /*j*/; return</lang>
- output when using the default inputs:
index │ first 50 cyclops numbers ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 0 101 102 103 104 105 106 107 108 109 11 │ 201 202 203 204 205 206 207 208 209 301 21 │ 302 303 304 305 306 307 308 309 401 402 31 │ 403 404 405 406 407 408 409 501 502 503 41 │ 504 505 506 507 508 509 601 602 603 604 ───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────── index │ first 50 prime cyclops numbers ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 101 103 107 109 307 401 409 503 509 601 11 │ 607 701 709 809 907 11,027 11,047 11,057 11,059 11,069 21 │ 11,071 11,083 11,087 11,093 12,011 12,037 12,041 12,043 12,049 12,071 31 │ 12,073 12,097 13,033 13,037 13,043 13,049 13,063 13,093 13,099 14,011 41 │ 14,029 14,033 14,051 14,057 14,071 14,081 14,083 14,087 15,013 15,017 ───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────── index │ first 50 blind prime cyclops numbers ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 101 103 107 109 307 401 503 509 601 607 11 │ 701 709 809 907 11,071 11,087 11,093 12,037 12,049 12,097 21 │ 13,099 14,029 14,033 14,051 14,071 14,081 14,083 14,087 15,031 15,053 31 │ 15,083 16,057 16,063 16,067 16,069 16,097 17,021 17,033 17,041 17,047 41 │ 17,053 17,077 18,047 18,061 18,077 18,089 19,013 19,031 19,051 19,073 ───────┴─────────────────────────────────────────────────────────────────────────────────────────────────────────────── index │ first 50 palindromic prime cyclops numbers ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 101 16,061 31,013 35,053 38,083 73,037 74,047 91,019 94,049 1,120,211 11 │ 1,150,511 1,160,611 1,180,811 1,190,911 1,250,521 1,280,821 1,360,631 1,390,931 1,490,941 1,520,251 21 │ 1,550,551 1,580,851 1,630,361 1,640,461 1,660,661 1,670,761 1,730,371 1,820,281 1,880,881 1,930,391 31 │ 1,970,791 3,140,413 3,160,613 3,260,623 3,310,133 3,380,833 3,460,643 3,470,743 3,590,953 3,670,763 41 │ 3,680,863 3,970,793 7,190,917 7,250,527 7,310,137 7,540,457 7,630,367 7,690,967 7,750,577 7,820,287 ───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────
Ruby
<lang ruby>require 'prime'
NONZEROS = %w(1 2 3 4 5 6 7 8 9)
cyclopes = Enumerator.new do |y|
(0..).each do |n|
NONZEROS.repeated_permutation(n) do |lside|
NONZEROS.repeated_permutation(n) do |rside|
y << (lside.join + "0" + rside.join).to_i
end
end
end
end
prime_cyclopes = Enumerator.new {|y| cyclopes.each {|c| y << c if c.prime?} } blind_prime_cyclopes = Enumerator.new {|y| prime_cyclopes.each {|c| y << c if c.to_s.delete("0").to_i.prime?} } palindromic_prime_cyclopes = Enumerator.new {|y| prime_cyclopes.each {|c| y << c if c.to_s == c.to_s.reverse} }
n, m = 50, 10_000_000 ["cyclopes", "prime cyclopes", "blind prime cyclopes", "palindromic prime cyclopes"].zip( [cyclopes, prime_cyclopes, blind_prime_cyclopes, palindromic_prime_cyclopes]).each do |name, enum|
cycl, idx = enum.each_with_index.detect{|n, i| n > m}
puts "The first #{n} #{name} are: \n#{enum.take(n).to_a}\nFirst #{name} term > #{m}: #{cycl} at index: #{idx}.", ""
end </lang>
- Output:
The first 50 cyclopes are: [0, 101, 102, 103, 104, 105, 106, 107, 108, 109, 201, 202, 203, 204, 205, 206, 207, 208, 209, 301, 302, 303, 304, 305, 306, 307, 308, 309, 401, 402, 403, 404, 405, 406, 407, 408, 409, 501, 502, 503, 504, 505, 506, 507, 508, 509, 601, 602, 603, 604] First cyclopes term > 10000000: 111101111 at index: 538084. The first 50 prime cyclopes are: [101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 11027, 11047, 11057, 11059, 11069, 11071, 11083, 11087, 11093, 12011, 12037, 12041, 12043, 12049, 12071, 12073, 12097, 13033, 13037, 13043, 13049, 13063, 13093, 13099, 14011, 14029, 14033, 14051, 14057, 14071, 14081, 14083, 14087, 15013, 15017] First prime cyclopes term > 10000000: 111101129 at index: 39319. The first 50 blind prime cyclopes are: [101, 103, 107, 109, 307, 401, 503, 509, 601, 607, 701, 709, 809, 907, 11071, 11087, 11093, 12037, 12049, 12097, 13099, 14029, 14033, 14051, 14071, 14081, 14083, 14087, 15031, 15053, 15083, 16057, 16063, 16067, 16069, 16097, 17021, 17033, 17041, 17047, 17053, 17077, 18047, 18061, 18077, 18089, 19013, 19031, 19051, 19073] First blind prime cyclopes term > 10000000: 111101161 at index: 11393. The first 50 palindromic prime cyclopes are: [101, 16061, 31013, 35053, 38083, 73037, 74047, 91019, 94049, 1120211, 1150511, 1160611, 1180811, 1190911, 1250521, 1280821, 1360631, 1390931, 1490941, 1520251, 1550551, 1580851, 1630361, 1640461, 1660661, 1670761, 1730371, 1820281, 1880881, 1930391, 1970791, 3140413, 3160613, 3260623, 3310133, 3380833, 3460643, 3470743, 3590953, 3670763, 3680863, 3970793, 7190917, 7250527, 7310137, 7540457, 7630367, 7690967, 7750577, 7820287] First palindromic prime cyclopes term > 10000000: 114808411 at index: 66.
Sidef
<lang ruby>func cyclops_numbers(base = 10) {
Enumerator({|callback|
var digits = @(1 .. base-1)
for k in (0 .. Inf `by` 2) {
digits.variations_with_repetition(k, {|*a|
a = (a.first(a.len>>1) + [0] + a.last(a.len>>1))
callback(a.flip.digits2num(base))
})
}
})
}
func palindromic_cyclops_numbers(base = 10) {
Enumerator({|callback|
var digits = @(1 .. base-1)
for k in (0..Inf) {
digits.variations_with_repetition(k, {|*a|
a = (a + [0] + a.flip)
callback(a.flip.digits2num(base))
})
}
})
}
func prime_cyclops(base = 10) {
var iter = cyclops_numbers(base)
Enumerator({|callback|
iter.each {|n|
callback(n) if n.is_prime
}
})
}
func blind_prime_cyclops(base = 10) {
var iter = prime_cyclops(base)
Enumerator({|callback|
iter.each {|n|
var k = (n.len(base)-1)>>1
var r = ipow(base, k)
if (r*idiv(n, r*base) + n%r -> is_prime) {
callback(n)
}
}
})
}
func palindromic_prime_cyclops(base = 10) {
var iter = palindromic_cyclops_numbers(base)
Enumerator({|callback|
iter.each {|n|
callback(n) if n.is_prime
}
})
}
for text,f in ([
[, cyclops_numbers], ['prime', prime_cyclops], ['blind prime', blind_prime_cyclops], ['palindromic prime', palindromic_prime_cyclops],
]) {
with (50) {|k|
say "First #{k} #{text} cyclops numbers:"
f().first(k).each_slice(10, {|*a|
a.map { '%7s' % _ }.join(' ').say
})
}
var min = 10_000_000
var iter = f()
var index = 0
var arr = Enumerator({|callback|
iter.each {|n|
callback([index, n]) if (n > min)
++index
}
}).first(1)[0]
say "\nFirst #{text} term > #{min.commify}: #{arr[1].commify} at (zero based) index: #{arr[0].commify}\n"
}</lang>
- Output:
First 50 cyclops numbers:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
First term > 10,000,000: 111,101,111 at (zero based) index: 538,084
First 50 prime cyclops numbers:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11027 11047 11057 11059 11069
11071 11083 11087 11093 12011 12037 12041 12043 12049 12071
12073 12097 13033 13037 13043 13049 13063 13093 13099 14011
14029 14033 14051 14057 14071 14081 14083 14087 15013 15017
First prime term > 10,000,000: 111,101,129 at (zero based) index: 39,319
First 50 blind prime cyclops numbers:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11071 11087 11093 12037 12049 12097
13099 14029 14033 14051 14071 14081 14083 14087 15031 15053
15083 16057 16063 16067 16069 16097 17021 17033 17041 17047
17053 17077 18047 18061 18077 18089 19013 19031 19051 19073
First blind prime term > 10,000,000: 111,101,161 at (zero based) index: 11,393
First 50 palindromic prime cyclops numbers:
101 16061 31013 35053 38083 73037 74047 91019 94049 1120211
1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251
1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391
1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763
3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287
First palindromic prime term > 10,000,000: 114,808,411 at (zero based) index: 66
Wren
<lang ecmascript>import "/math" for Int import "/seq" for Lst import "/fmt" for Fmt import "/str" for Str
var findFirst = Fn.new { |list|
var i = 0
for (n in list) {
if (n > 1e7) return [n, i]
i = i + 1
}
}
var ranges = [0..0, 101..909, 11011..99099, 1110111..9990999, 111101111..119101111] var cyclops = [] for (r in ranges) {
var numDigits = r.from.toString.count
var center = (numDigits / 2).floor
for (i in r) {
var digits = Int.digits(i)
if (digits[center] == 0 && digits.count { |d| d == 0 } == 1) cyclops.add(i)
}
}
System.print("The first 50 cyclops numbers are:") var candidates = cyclops[0...50] var ni = findFirst.call(cyclops) for (chunk in Lst.chunks(candidates, 10)) Fmt.print("$,6d", chunk) Fmt.print("\nFirst such number > 10 million is $,d at zero-based index $,d", ni[0], ni[1])
System.print("\n\nThe first 50 prime cyclops numbers are:") var primes = cyclops.where { |n| Int.isPrime(n) } candidates = primes.take(50).toList ni = findFirst.call(primes) for (chunk in Lst.chunks(candidates, 10)) Fmt.print("$,6d", chunk) Fmt.print("\nFirst such number > 10 million is $,d at zero-based index $,d", ni[0], ni[1])
System.print("\n\nThe first 50 blind prime cyclops numbers are:") var bpcyclops = [] var ppcyclops = [] for (p in primes) {
var ps = p.toString var numDigits = ps.count var center = (numDigits/2).floor var noMiddle = Num.fromString(Str.delete(ps, center)) if (Int.isPrime(noMiddle)) bpcyclops.add(p) if (ps == ps[-1..0]) ppcyclops.add(p)
} candidates = bpcyclops[0...50] ni = findFirst.call(bpcyclops) for (chunk in Lst.chunks(candidates, 10)) Fmt.print("$,6d", chunk) Fmt.print("\nFirst such number > 10 million is $,d at zero-based index $,d", ni[0], ni[1])
System.print("\n\nThe first 50 palindromic prime cyclops numbers are:") candidates = ppcyclops[0...50] ni = findFirst.call(ppcyclops) for (chunk in Lst.chunks(candidates, 8)) Fmt.print("$,9d", chunk) Fmt.print("\nFirst such number > 10 million is $,d at zero-based index $,d", ni[0], ni[1])</lang>
- Output:
The first 50 cyclops numbers are:
0 101 102 103 104 105 106 107 108 109
201 202 203 204 205 206 207 208 209 301
302 303 304 305 306 307 308 309 401 402
403 404 405 406 407 408 409 501 502 503
504 505 506 507 508 509 601 602 603 604
First such number > 10 million is 111,101,111 at zero-based index 538,084
The first 50 prime cyclops numbers are:
101 103 107 109 307 401 409 503 509 601
607 701 709 809 907 11,027 11,047 11,057 11,059 11,069
11,071 11,083 11,087 11,093 12,011 12,037 12,041 12,043 12,049 12,071
12,073 12,097 13,033 13,037 13,043 13,049 13,063 13,093 13,099 14,011
14,029 14,033 14,051 14,057 14,071 14,081 14,083 14,087 15,013 15,017
First such number > 10 million is 111,101,129 at zero-based index 39,319
The first 50 blind prime cyclops numbers are:
101 103 107 109 307 401 503 509 601 607
701 709 809 907 11,071 11,087 11,093 12,037 12,049 12,097
13,099 14,029 14,033 14,051 14,071 14,081 14,083 14,087 15,031 15,053
15,083 16,057 16,063 16,067 16,069 16,097 17,021 17,033 17,041 17,047
17,053 17,077 18,047 18,061 18,077 18,089 19,013 19,031 19,051 19,073
First such number > 10 million is 111,101,161 at zero-based index 11,393
The first 50 palindromic prime cyclops numbers are:
101 16,061 31,013 35,053 38,083 73,037 74,047 91,019
94,049 1,120,211 1,150,511 1,160,611 1,180,811 1,190,911 1,250,521 1,280,821
1,360,631 1,390,931 1,490,941 1,520,251 1,550,551 1,580,851 1,630,361 1,640,461
1,660,661 1,670,761 1,730,371 1,820,281 1,880,881 1,930,391 1,970,791 3,140,413
3,160,613 3,260,623 3,310,133 3,380,833 3,460,643 3,470,743 3,590,953 3,670,763
3,680,863 3,970,793 7,190,917 7,250,527 7,310,137 7,540,457 7,630,367 7,690,967
7,750,577 7,820,287
First such number > 10 million is 114,808,411 at zero-based index 66
XPL0
<lang XPL0>func IsCyclops(N); \Return 'true' if N is a cyclops number int N, I, J, K; char A(9); [I:= 0; \parse digits into array A repeat N:= N/10;
A(I):= rem(0);
I:= I+1;
until N=0; if (I&1) = 0 then return false; \must have odd number of digits K:= I>>1; if A(K) # 0 then return false; \middle digit must be 0 for J:= 0 to I-1 do \other digits must not be 0
if A(J)=0 & J#K then return false;
return true; ];
func IsPrime(N); \Return 'true' if N > 2 is a prime number int N, I; [if (N&1) = 0 \even number\ then return false; for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false; I:= I+1; ];
return true; ];
func Blind(N); \Return blinded cyclops number int N, I, J, K; \i.e. center zero removed char A(9); [I:= 0; \parse digits into array A repeat N:= N/10;
A(I):= rem(0);
I:= I+1;
until N=0; N:= A(I-1); \most significant digit K:= I>>1; for J:= I-2 downto 0 do
if J#K then \skip middle zero
N:= N*10 + A(J);
return N; ];
func Reverse(N); \Return N with its digits reversed int N, M; [M:= 0; repeat N:= N/10;
M:= M*10 + rem(0);
until N=0; return M; ];
func IntLen(N); \Return number of decimal digits in N int N; int P, I; [P:= 10; for I:= 1 to 9 do \assumes N is 32-bits
[if P>N then return I;
P:= P*10;
];
return 10; ];
int Count, N;
func Show; \Show results and return 'true' when done [Count:= Count+1;
if Count <= 50 then
[IntOut(0, N);
if rem(Count/10) = 0 then CrLf(0) else ChOut(0, 9\tab\);
];
if N > 10_000_000 then
[Text(0, "First such number > 10,000,000: ");
IntOut(0, N);
Text(0, " at zero based index: ");
IntOut(0, Count-1);
CrLf(0);
return true;
];
return false; ];
proc Common(Filter); \Common code gathered here int Filter; [Count:= 0; N:= 0; loop [if IsCyclops(N) then
case Filter of
0: if Show then quit;
1: if IsPrime(N) then
if Show then quit;
2: if IsPrime(N) then if IsPrime(Blind(N)) then
if Show then quit;
3: if N=Reverse(N) then if IsPrime(N) then
if Show then quit
other [];
N:= N+1;
if (IntLen(N)&1) = 0 then N:= N*10; \must have odd number of digits
];
];
[Text(0, "First 50 cyclops numbers: "); Common(0); Text(0, " First 50 prime cyclops numbers: "); Common(1); Text(0, " First 50 blind prime cyclops numbers: "); Common(2); Text(0, " First 50 palindromic prime cyclops numbers: "); Common(3); ]</lang>
- Output:
First 50 cyclops numbers: 0 101 102 103 104 105 106 107 108 109 201 202 203 204 205 206 207 208 209 301 302 303 304 305 306 307 308 309 401 402 403 404 405 406 407 408 409 501 502 503 504 505 506 507 508 509 601 602 603 604 First such number > 10,000,000: 111101111 at zero based index: 538084 First 50 prime cyclops numbers: 101 103 107 109 307 401 409 503 509 601 607 701 709 809 907 11027 11047 11057 11059 11069 11071 11083 11087 11093 12011 12037 12041 12043 12049 12071 12073 12097 13033 13037 13043 13049 13063 13093 13099 14011 14029 14033 14051 14057 14071 14081 14083 14087 15013 15017 First such number > 10,000,000: 111101129 at zero based index: 39319 First 50 blind prime cyclops numbers: 101 103 107 109 307 401 503 509 601 607 701 709 809 907 11071 11087 11093 12037 12049 12097 13099 14029 14033 14051 14071 14081 14083 14087 15031 15053 15083 16057 16063 16067 16069 16097 17021 17033 17041 17047 17053 17077 18047 18061 18077 18089 19013 19031 19051 19073 First such number > 10,000,000: 111101161 at zero based index: 11393 First 50 palindromic prime cyclops numbers: 101 16061 31013 35053 38083 73037 74047 91019 94049 1120211 1150511 1160611 1180811 1190911 1250521 1280821 1360631 1390931 1490941 1520251 1550551 1580851 1630361 1640461 1660661 1670761 1730371 1820281 1880881 1930391 1970791 3140413 3160613 3260623 3310133 3380833 3460643 3470743 3590953 3670763 3680863 3970793 7190917 7250527 7310137 7540457 7630367 7690967 7750577 7820287 First such number > 10,000,000: 114808411 at zero based index: 66