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Cyclops numbers

From Rosetta Code
Cyclops numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

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[edit]

Generates the sequence.

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
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[edit]

 
# 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")
}
 
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#[edit]

This task uses Extensible Prime Generator (F#)

Cyclops function[edit]

 
// 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
 

The tasks[edit]

First 50 Cyclop numbers
 
cyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn ""
 
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
 
primeCyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn ""
 
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
 
blindCyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn ""
 
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
 
palindromicCyclops|>Seq.take 50|>Seq.iter(printf "%d "); printfn ""
 
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
 
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
 
Output:
First Cyclop number > 10,000,000 is 111101111 at index 538084
First prime Cyclop number > 10,000,000
 
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
 
Output:
First prime Cyclop number > 10,000,000 is 111101129 at index 39319
First blind Cyclop number > 10,000,000
 
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
 
Output:
First blind Cyclop number > 10,000,000 is 111101161 at index 11393
First palindromic prime Cyclop number > 10,000,000
 
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
 
Output:
First palindromic prime Cyclop number > 10,000,000 is 114808411 at index 66

Factor[edit]

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 + 1 to both sides, adding 1 to the number of digits, and setting the new max as max := 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.


Works with: Factor version 0.99 2021-06-02
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 ] [email protected] ;
 
 
 
! ====================={[ 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
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[edit]

Translation of: Wren
Library: Go-rcu
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)
}
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[edit]

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]
]
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[edit]

Note that the indices shown in the output are 0-based, as jq's index origin is 0.

 
## 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) ;
 

The tasks

 
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))
 
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[edit]

Julia's indexes are 1 based, hence one greater than those of 0 based indices.

 
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()
 
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.

Nim[edit]

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."
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[edit]

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

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.
 
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[edit]

Translation of: Raku
Library: ntheory
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;
}
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[edit]

Library: Phix/online

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[edit]

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[edit]

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;
}
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[edit]

/*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 [email protected].#+2 by 2 for max(0, hip%[email protected].#%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
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[edit]

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
 
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[edit]

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"
}
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[edit]

Library: Wren-seq
Library: Wren-fmt
Library: Wren-str
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])
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[edit]

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);
]
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