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# Extra primes

Extra primes 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.
Definition

n   is an   extra prime   if   n   is prime and its decimal digits and sum of digits are also primes.

Show the extra primes under   10,000.

Reference

OEIS:A062088 - Primes with every digit a prime and the sum of the digits a prime.

with Ada.Text_Io; procedure Extra_Primes is    type Number   is new Long_Integer range 0 .. Long_Integer'Last;    package Number_Io is new Ada.Text_Io.Integer_Io (Number);    function Is_Prime (A : Number) return Boolean is      D : Number;   begin      if A < 2       then return False; end if;      if A in 2 .. 3 then return True;  end if;      if A mod 2 = 0 then return False; end if;      if A mod 3 = 0 then return False; end if;      D := 5;      while D * D <= A loop         if A mod D = 0 then            return False;         end if;         D := D + 2;         if A mod D = 0 then            return False;         end if;         D := D + 4;      end loop;      return True;   end Is_Prime;    subtype Digit is Number range 0 .. 9;   type Digit_Array is array (Positive range <>) of Digit;    function To_Digits (N : Number) return Digit_Array is      Image : constant String := Number'Image (N);      Res   : Digit_Array (2 .. Image'Last);   begin      for A in Image'First + 1 .. Image'Last loop         Res (A) := Character'Pos (Image (A)) - Character'Pos ('0');      end loop;      return Res;   end To_Digits;    function All_Prime (Dig : Digit_Array) return Boolean is     (for all D of Dig => Is_Prime (D));    function Sum_Of (Dig : Digit_Array) return Number is      Sum : Number := 0;   begin      for D of Dig loop         Sum := Sum + D;      end loop;      return Sum;   end Sum_Of;    use Ada.Text_Io;   Count : Natural := 0;begin   for N in Number range 1 .. 9_999 loop      if Is_Prime (N) then         declare            Dig : constant Digit_Array := To_Digits (N);         begin            if All_Prime (Dig) and Is_Prime (Sum_Of (Dig)) then               Count := Count + 1;               Number_Io.Put (N, Width => 4); Put ("  ");               if Count mod 8 = 0 then                  New_Line;               end if;            end if;         end;      end if;   end loop;   New_Line;   Put_Line (Count'Image & " extra primes.");end Extra_Primes;
Output:
   2     3     5     7    23   223   227   337
353   373   557   577   733   757   773  2333
2357  2377  2557  2753  2777  3253  3257  3323
3527  3727  5233  5237  5273  5323  5527  7237
7253  7523  7723  7727
36 extra primes.

## ALGOL W

As the digits can only be 2, 3, 5 or 7 (see the Wren sample) we can easily generate the candidates for the sequence.

begin    % find extra primes - numbers whose digits are prime and whose   %    % digit sum is prime                                             %    % the digits can only be 2, 3, 5, 7                              %    % as we are looking for extra primes below 10 000, the maximum   %    % number to consider is 7 777, whose digit sum is 28             %    integer MAX_PRIME;    MAX_PRIME := 7777;    begin        logical array isPrime ( 1 :: MAX_PRIME );        integer numberCount;        % sieve the primes up to MAX_PRIME                           %        for i := 1 until MAX_PRIME do isPrime ( i ) := true;        isPrime( 1 ) := false;        for i := 2 until truncate( sqrt( MAX_PRIME ) ) do begin            if isPrime ( i ) then for p := i * i step i until MAX_PRIME do isPrime( p ) := false        end for_i ;        % find the extra primes                                      %        numberCount := 0;        write();        for d1 := 0, 2, 3, 5, 7 do begin            for d2 := 0, 2, 3, 5, 7 do begin                if d2 not = 0 or d1 = 0 then begin                    for d3 := 0, 2, 3, 5, 7 do begin                        if d3 not = 0 or ( d1 = 0 and d2 = 0 ) then begin                            for d4 := 2, 3, 5, 7 do begin                                integer sum, n;                                n := 0;                                for d := d1, d2, d3, d4 do n := ( n * 10 ) + d;                                sum := d1 + d2 + d3 + d4;                                if isPrime( sum ) and isPrime( n ) then begin                                    % found a prime whose prime      %                                    % digits sum to a prime          %                                    writeon( i_w := 5, s_w := 1, n );                                    numberCount := numberCount + 1;                                    if numberCount rem 12 = 0 then write()                                end if_isPrime_sum                            end for_d4                        end if_d3_ne_0_or_d1_eq_0_and_d2_e_0                    end for_d3                end if_d2_ne_0_or_d1_eq_0            end for_d2        end for_d1    endend.
Output:
    2     3     5     7    23   223   227   337   353   373   557   577
733   757   773  2333  2357  2377  2557  2753  2777  3253  3257  3323
3527  3727  5233  5237  5273  5323  5527  7237  7253  7523  7723  7727


## APL

extraPrimes←{    pd←0 2 3 5 7    ds←↓⍉(∧⌿ds∊pd)/ds←10(⊥⍣¯1)1↓⍳⍵    ds←↑((∧/2(≤≥0=⊢)/⊢)¨ds)/ds    ns←(ns≤⍵)/ns←10⊥⍉ds    ss←+/(⍴ns)↑ds    sieve←~(1+⌈/ns,ss){        r←1↓⍺⍴(⍺⌊⍵)↑1        ∨/r:(r∧⍵≠⍳⍺-1)∨⍺∇1+2*r⍳1        (⍺-1)/0    }2    (sieve[ns]∧sieve[ss])/ns}
Output:
      extraPrimes 100002 3 5 7 23 27 223 227 333 337 353 373 377 533 553 557 577 733 737 757      773 777 2223 2227 2333 2353 2357 2377 2533 2537 2557 2573 2577      2737 2753 2757 2773 2777 3233 3253 3257 3277 3323 3523 3527 3727      5233 5237 5257 5273 5277 5323 5327 5527 5723 5727 7237 7253 7257      7273 7277 7327 7523 7527 7723 7727

## AWK

 # syntax: GAWK -f EXTRA_PRIMES.AWKBEGIN {    for (i=1; i<10000; i++) {      if (is_prime(i)) {        sum = fail = 0        for (j=1; j<=length(i); j++) {          sum += n = substr(i,j,1)          if (!is_prime(n)) {            fail = 1            break          }        }        if (is_prime(sum) && fail == 0) {          printf("%2d %4d\n",++count,i)        }      }    }    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)}
Output:
 1    2
2    3
3    5
4    7
5   23
6  223
7  227
8  337
9  353
10  373
11  557
12  577
13  733
14  757
15  773
16 2333
17 2357
18 2377
19 2557
20 2753
21 2777
22 3253
23 3257
24 3323
25 3527
26 3727
27 5233
28 5237
29 5273
30 5323
31 5527
32 7237
33 7253
34 7523
35 7723
36 7727


## BASIC

10 DEFINT A-Z: DIM S(7777),D(4): DATA 0,2,3,5,715 FOR I=0 TO 4: READ D(I): NEXT20 FOR I=2 TO SQR(7777)30 FOR J=I*I TO 7777 STEP I: S(J)=-1: NEXT40 NEXT50 FOR A=0 TO 460 FOR B=0 TO 4: IF A<>0 AND B=0 THEN 13070 FOR C=0 TO 4: IF B<>0 AND C=0 THEN 12080 FOR D=1 TO 490 I=D(A)*1000 + D(B)*100 + D(C)*10 + D(D)95 S=D(A) + D(B) + D(C) + D(D)100 IF NOT (S(S) OR S(I)) THEN PRINT I,110 NEXT120 NEXT130 NEXT140 NEXT
Output:
 2             3             5             7             23
223           227           337           353           373
557           577           733           757           773
2333          2357          2377          2557          2753
2777          3253          3257          3323          3527
3727          5233          5237          5273          5323
5527          7237          7253          7523          7723
7727

## C

#include <locale.h>#include <stdbool.h>#include <stdio.h> unsigned int next_prime_digit_number(unsigned int n) {    if (n == 0)        return 2;    switch (n % 10) {    case 2:        return n + 1;    case 3:    case 5:        return n + 2;    default:        return 2 + next_prime_digit_number(n/10) * 10;    }} bool is_prime(unsigned int n) {    if (n < 2)        return false;    if ((n & 1) == 0)        return n == 2;    if (n % 3 == 0)        return n == 3;    if (n % 5 == 0)        return n == 5;    static const unsigned int wheel[] = { 4,2,4,2,4,6,2,6 };    unsigned int p = 7;    for (;;) {        for (int w = 0; w < sizeof(wheel)/sizeof(wheel[0]); ++w) {            if (p * p > n)                return true;            if (n % p == 0)                return false;            p += wheel[w];        }    }} unsigned int digit_sum(unsigned int n) {    unsigned int sum = 0;    for (; n > 0; n /= 10)        sum += n % 10;    return sum;} int main() {    setlocale(LC_ALL, "");    const unsigned int limit1 = 10000;    const unsigned int limit2 = 1000000000;    const int last = 10;    unsigned int p = 0, n = 0;    unsigned int extra_primes[last];    printf("Extra primes under %'u:\n", limit1);    while ((p = next_prime_digit_number(p)) < limit2) {        if (is_prime(digit_sum(p)) && is_prime(p)) {            ++n;            if (p < limit1)                printf("%2u: %'u\n", n, p);            extra_primes[n % last] = p;        }    }    printf("\nLast %d extra primes under %'u:\n", last, limit2);    for (int i = last - 1; i >= 0; --i)        printf("%'u: %'u\n", n-i, extra_primes[(n-i) % last]);    return 0;}
Output:
Extra primes under 10,000:
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2,333
17: 2,357
18: 2,377
19: 2,557
20: 2,753
21: 2,777
22: 3,253
23: 3,257
24: 3,323
25: 3,527
26: 3,727
27: 5,233
28: 5,237
29: 5,273
30: 5,323
31: 5,527
32: 7,237
33: 7,253
34: 7,523
35: 7,723
36: 7,727

Last 10 extra primes under 1,000,000,000:
9,049: 777,753,773
9,050: 777,755,753
9,051: 777,773,333
9,052: 777,773,753
9,053: 777,775,373
9,054: 777,775,553
9,055: 777,775,577
9,056: 777,777,227
9,057: 777,777,577
9,058: 777,777,773


## C++

#include <iomanip>#include <iostream> unsigned int next_prime_digit_number(unsigned int n) {    if (n == 0)        return 2;    switch (n % 10) {    case 2:        return n + 1;    case 3:    case 5:        return n + 2;    default:        return 2 + next_prime_digit_number(n/10) * 10;    }} bool is_prime(unsigned int n) {    if (n < 2)        return false;    if ((n & 1) == 0)        return n == 2;    if (n % 3 == 0)        return n == 3;    if (n % 5 == 0)        return n == 5;    static constexpr unsigned int wheel[] = { 4,2,4,2,4,6,2,6 };    unsigned int p = 7;    for (;;) {        for (unsigned int w : wheel) {            if (p * p > n)                return true;            if (n % p == 0)                return false;            p += w;        }    }} unsigned int digit_sum(unsigned int n) {    unsigned int sum = 0;    for (; n > 0; n /= 10)        sum += n % 10;    return sum;} int main() {    std::cout.imbue(std::locale(""));    const unsigned int limit1 = 10000;    const unsigned int limit2 = 1000000000;    const int last = 10;    unsigned int p = 0, n = 0;    unsigned int extra_primes[last];    std::cout << "Extra primes under " << limit1 << ":\n";    while ((p = next_prime_digit_number(p)) < limit2) {        if (is_prime(digit_sum(p)) && is_prime(p)) {            ++n;            if (p < limit1)                std::cout << std::setw(2) << n << ": " << p << '\n';            extra_primes[n % last] = p;        }    }    std::cout << "\nLast " << last << " extra primes under " << limit2 << ":\n";    for (int i = last - 1; i >= 0; --i)        std::cout << n-i << ": " << extra_primes[(n-i) % last] << '\n';    return 0;}
Output:
Extra primes under 10,000:
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2,333
17: 2,357
18: 2,377
19: 2,557
20: 2,753
21: 2,777
22: 3,253
23: 3,257
24: 3,323
25: 3,527
26: 3,727
27: 5,233
28: 5,237
29: 5,273
30: 5,323
31: 5,527
32: 7,237
33: 7,253
34: 7,523
35: 7,723
36: 7,727

Last 10 extra primes under 1,000,000,000:
9,049: 777,753,773
9,050: 777,755,753
9,051: 777,773,333
9,052: 777,773,753
9,053: 777,775,373
9,054: 777,775,553
9,055: 777,775,577
9,056: 777,777,227
9,057: 777,777,577
9,058: 777,777,773


## Cowgol

include "cowgol.coh";const MAXPRIME := 7777; var sieve: uint8[MAXPRIME+1];MemZero(&sieve[0], @bytesof sieve);typedef Candidate is @indexof sieve;var cand: Candidate := 2;loop    var mark := cand * cand;    if mark > MAXPRIME then break; end if;    while mark <= MAXPRIME loop        sieve[mark] := 1;        mark := mark + cand;    end loop;    cand := cand + 1;end loop; var digits: Candidate[] := {0, 2, 3, 5, 7};var i1: uint8;var i2: uint8;var i3: uint8;var i4: uint8;i1 := 0;while i1 < 5 loop    i2 := 0;    while i2 < 5 loop        if i1 == 0 or i2 != 0 then            i3 := 0;            while i3 < 5 loop                if i2 == 0 or i3 != 0 then                    i4 := 1;                    while i4 < 5 loop                                        cand := digits[i1] * 1000                              + digits[i2] * 100                              + digits[i3] * 10                              + digits[i4];                        var sum := digits[i1]                                 + digits[i2]                                 + digits[i3]                                 + digits[i4];                        if sieve[cand] | sieve[sum] == 0 then                            print_i32(cand as uint32);                            print_nl();                        end if;                        i4 := i4 + 1;                    end loop;                end if;                i3 := i3 + 1;            end loop;        end if;        i2 := i2 + 1;    end loop;    i1 := i1 + 1;end loop;
Output:
2
3
5
7
23
223
227
337
353
373
557
577
733
757
773
2333
2357
2377
2557
2753
2777
3253
3257
3323
3527
3727
5233
5237
5273
5323
5527
7237
7253
7523
7723
7727

## D

Translation of: Java
import std.stdio; int nextPrimeDigitNumber(int n) {    if (n == 0) {        return 2;    }    switch (n % 10) {        case 2:            return n + 1;        case 3:        case 5:            return n + 2;        default:            return 2 + nextPrimeDigitNumber(n / 10) * 10;    }} bool isPrime(int n) {    if (n < 2) {        return false;    }    if ((n & 1) == 0) {        return n == 2;    }    if (n % 3 == 0) {        return n == 3;    }    if (n % 5 == 0) {        return n == 5;    }     int p = 7;    while (true) {        foreach (w; [4, 2, 4, 2, 4, 6, 2, 6]) {            if (p * p > n) {                return true;            }            if (n % p == 0) {                return false;            }            p += w;        }    }} int digitSum(int n) {    int sum = 0;    for (; n > 0; n /= 10) {        sum += n % 10;    }    return sum;} void main() {    immutable limit = 10_000;    int p = 0;    int n = 0;     writeln("Extra primes under ", limit);    while (p < limit) {        p = nextPrimeDigitNumber(p);        if (isPrime(p) && isPrime(digitSum(p))) {            n++;            writefln("%2d: %d", n, p);        }    }    writeln;}
Output:
Extra primes under 10000
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727

## F#

### The Function

 // Extra Primes. Nigel Galloway: January 9th., 2021let izXprime g=let rec fN n g=match n with 0L->isPrime64 g |_->if isPrime64(n%10L) then fN (n/10L) (n%10L+g) else false in fN g 0L

Extra primes below 10,000
 primes64() |> Seq.filter izXprime |> Seq.takeWhile((>) 10000L) |> Seq.iteri(printfn "%3d->%d")
Output:
  0->2
1->3
2->5
3->7
4->23
5->223
6->227
7->337
8->353
9->373
10->557
11->577
12->733
13->757
14->773
15->2333
16->2357
17->2377
18->2557
19->2753
20->2777
21->3253
22->3257
23->3323
24->3527
25->3727
26->5233
27->5237
28->5273
29->5323
30->5527
31->7237
32->7253
33->7523
34->7723
35->7727

Last 10 Extra primes below 1,000,000,000
 primes64()|>Seq.takeWhile((>)1000000000L)|>Seq.rev|>Seq.filter izXprime|>Seq.take 10|>Seq.rev|>Seq.iter(printf "%d ");printfn ""
Output:
777753773 777755753 777773333 777773753 777775373 777775553 777775577 777777227 777777577 777777773

Last 10 Extra primes below 10,000,000,000
 primes64()|>Seq.skipWhile((>)7770000000L)|>Seq.takeWhile((>)7777777777L)|>List.ofSeq|>List.filter izXprime|>List.rev|>List.take 10|>List.rev|>List.iter(printf "%d ");printfn ""
Output:
7777733273 7777737727 7777752737 7777753253 7777772773 7777773257 7777773277 7777775273 7777777237 7777777327


## Factor

Translation of: Wren
Works with: Factor version 0.99 2020-08-14
USING: formatting io kernel math math.functions math.primessequences sequences.extras ; : digit ( seq seq -- seq ) [ suffix ] cartesian-map concat ;: front ( -- seq ) { { 2 } { 3 } { 5 } { 7 } } ;: middle ( seq -- newseq ) { 2 3 5 7 } digit ;: end ( seq -- newseq ) { 3 7 } digit ; : candidates ( -- seq )    front    front end    front middle end    front middle middle end    append append append ; : digits>number ( seq -- n )    <reversed> 0 [ 10^ * + ] reduce-index ; "The extra primes with up to 4 digits are:" printcandidates[ sum prime? ] filter[ digits>number ] [ prime? ] map-filter[ 1 + swap "%2d: %4d\n" printf ] each-index
Output:
The extra primes with up to 4 digits are:
1:    2
2:    3
3:    5
4:    7
5:   23
6:  223
7:  227
8:  337
9:  353
10:  373
11:  557
12:  577
13:  733
14:  757
15:  773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727


## Forth

: is_prime? ( n -- flag )  dup 2 < if drop false exit then  dup 2 mod 0= if 2 = exit then  dup 3 mod 0= if 3 = exit then  5  begin    2dup dup * >=  while    2dup mod 0= if 2drop false exit then    2 +    2dup mod 0= if 2drop false exit then    4 +  repeat  2drop true ; : next_prime_digit_number ( n -- n )  dup 0= if drop 2 exit then  dup 10 mod  dup 2 = if drop 1+ exit then  dup 3 = if drop 2 + exit then  5 = if 2 + exit then  10 / recurse 10 * 2 + ; : digit_sum ( u -- u )  dup 10 < if exit then  10 /mod recurse + ; : next_extra_prime ( n -- n )  begin    next_prime_digit_number    dup digit_sum is_prime? if      dup is_prime?    else false then  until ; : print_extra_primes ( n -- )  0  begin    next_extra_prime 2dup >  while    dup . cr  repeat  2drop ; : count_extra_primes ( n -- n )  0 0 >r  begin    next_extra_prime 2dup >  while    r> 1+ >r  repeat  2drop r> ; ." Extra primes under 10000:" cr10000 print_extra_primes 100000000 count_extra_primes." Number of extra primes under 100000000: " . cr bye
Output:
Extra primes under 10000:
2
3
5
7
23
223
227
337
353
373
557
577
733
757
773
2333
2357
2377
2557
2753
2777
3253
3257
3323
3527
3727
5233
5237
5273
5323
5527
7237
7253
7523
7723
7727
Number of extra primes under 100000000: 2498


## FreeBASIC

 dim as uinteger p(0 to 4) = {0,2,3,5,7}, d3, d2, d1, d0, pd1, pd2, pd3, pd0 function isprime( n as uinteger ) as boolean    if n mod 2 = 0 then return false    for i as uinteger = 3 to int(sqr(n))+1 step 2        if n mod i = 0 then return false    next i    return trueend function print "0002"  'special casefor d3 = 0 to 4    pd3 = p(d3)    for d2 = 0 to 4        if d3 > 0 and d2 = 0 then continue for        pd2 = p(d2)        for d1 = 0 to 4            if d2+d3 > 0 and d1 = 0 then continue for            pd1 = p(d1)            for d0 = 2 to 4                pd0 = p(d0)                if isprime(pd0 + 10*pd1 + 100*pd2 + 1000*pd3 ) and isprime( pd0 + pd1 + pd2 + pd3) then print pd3;pd2;pd1;pd0            next d0        next d1    next d2next d3
Output:
0002
0003
0005
0007
0023
0223
0227
0337
0353
0373
0557
0577
0733
0757
0773
2333
2357
2377
2557
2753
2777
3253
3257
3323
3527
3727
5233
5237
5273
5323
5527
7237
7253
7523
7723
7727

## Go

Translation of: Wren
package main import "fmt" func isPrime(n int) bool {    if n < 2 {        return false    }    if n%2 == 0 {        return n == 2    }    if n%3 == 0 {        return n == 3    }    d := 5    for d*d <= n {        if n%d == 0 {            return false        }        d += 2        if n%d == 0 {            return false        }        d += 4    }    return true} func main() {    digits := [4]int{2, 3, 5, 7}                      // the only digits which are primes    digits2 := [2]int{3, 7}                           // a prime > 5 can't end in 2 or 5    cands := [][2]int{{2, 2}, {3, 3}, {5, 5}, {7, 7}} // {number, digits sum}     for _, a := range digits {        for _, b := range digits2 {            cands = append(cands, [2]int{10*a + b, a + b})        }    }     for _, a := range digits {        for _, b := range digits {            for _, c := range digits2 {                cands = append(cands, [2]int{100*a + 10*b + c, a + b + c})            }        }    }     for _, a := range digits {        for _, b := range digits {            for _, c := range digits {                for _, d := range digits2 {                    cands = append(cands, [2]int{1000*a + 100*b + 10*c + d, a + b + c + d})                }            }        }    }     fmt.Println("The extra primes under 10,000 are:")    count := 0    for _, cand := range cands {        if isPrime(cand[0]) && isPrime(cand[1]) {            count++            fmt.Printf("%2d: %4d\n", count, cand[0])        }    }}
Output:
The extra primes under 10,000 are:
1:    2
2:    3
3:    5
4:    7
5:   23
6:  223
7:  227
8:  337
9:  353
10:  373
11:  557
12:  577
13:  733
14:  757
15:  773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727


## J

exprimes =: (] #~ *./@(1&p:)@(+/ , ])@(10 #.^:_1 ])"0)@(i.&.(p:^:_1))
Output:
   exprimes 100002 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727

## Java

Translation of: Go
public class ExtraPrimes {    private static int nextPrimeDigitNumber(int n) {        if (n == 0) {            return 2;        }        switch (n % 10) {            case 2:                return n + 1;            case 3:            case 5:                return n + 2;            default:                return 2 + nextPrimeDigitNumber(n / 10) * 10;        }    }     private static boolean isPrime(int n) {        if (n < 2) {            return false;        }        if ((n & 1) == 0) {            return n == 2;        }        if (n % 3 == 0) {            return n == 3;        }        if (n % 5 == 0) {            return n == 5;        }         int[] wheel = new int[]{4, 2, 4, 2, 4, 6, 2, 6};        int p = 7;        while (true) {            for (int w : wheel) {                if (p * p > n) {                    return true;                }                if (n % p == 0) {                    return false;                }                p += w;            }        }    }     private static int digitSum(int n) {        int sum = 0;        for (; n > 0; n /= 10) {            sum += n % 10;        }        return sum;    }     public static void main(String[] args) {        final int limit = 10_000;        int p = 0, n = 0;         System.out.printf("Extra primes under %d:\n", limit);        while (p < limit) {            p = nextPrimeDigitNumber(p);            if (isPrime(p) && isPrime(digitSum(p))) {                n++;                System.out.printf("%2d: %d\n", n, p);            }        }        System.out.println();    }}
Output:
Extra primes under 10000:
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727

## jq

Works with: jq

Works with gojq, the Go implementation of jq

For the definition of is_prime used here, see https://rosettacode.org/wiki/Additive_primes

One small point of interest is the declaration of $p before the inner function that references it.  # Input: the maximum width# Output: a streamdef extraprimes: [2,3,5,7] as$p  # Input: width  # Output: a stream of arrays of length $n drawn from$p  | def wide: . as $n | if . == 0 then [] else$p[] | [.] + (($n-1)|wide) end; range(1;.+1) as$maxlen  | ($maxlen | wide) | select( add | is_prime) | join("") | tonumber | select(is_prime) ; # The task: 4|extraprimes Output: 2 3 5 7 23 ... 7253 7523 7723 7727  ## Julia using Primes function extraprimes(maxlen) for i in 1:maxlen, combo in Iterators.product([[2, 3, 5, 7] for _ in 1:i]...) if isprime(sum(combo)) n = evalpoly(10, combo) isprime(n) && println(n) end endend extraprimes(4)  Output: 2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727  ## Kotlin Translation of: Java private fun nextPrimeDigitNumber(n: Int): Int { return if (n == 0) { 2 } else when (n % 10) { 2 -> n + 1 3, 5 -> n + 2 else -> 2 + nextPrimeDigitNumber(n / 10) * 10 }} private fun isPrime(n: Int): Boolean { if (n < 2) { return false } if (n and 1 == 0) { return n == 2 } if (n % 3 == 0) { return n == 3 } if (n % 5 == 0) { return n == 5 } val wheel = intArrayOf(4, 2, 4, 2, 4, 6, 2, 6) var p = 7 while (true) { for (w in wheel) { if (p * p > n) { return true } if (n % p == 0) { return false } p += w } }} private fun digitSum(n: Int): Int { var nn = n var sum = 0 while (nn > 0) { sum += nn % 10 nn /= 10 } return sum} fun main() { val limit = 10000 var p = 0 var n = 0 println("Extra primes under$limit:")    while (p < limit) {        p = nextPrimeDigitNumber(p)        if (isPrime(p) && isPrime(digitSum(p))) {            n++            println("%2d: %d".format(n, p))        }    }    println()}
Output:
Extra primes under 10000:
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727

## Lua

Translation of: C
function next_prime_digit_number(n)    if n == 0 then        return 2    end    local r = n % 10    if r == 2 then        return n + 1    end    if r == 3 or r == 5 then        return n + 2    end    return 2 + next_prime_digit_number(math.floor(n / 10)) * 10end function is_prime(n)    if n < 2 then        return false    end     if n % 2 == 0 then        return n == 2    end    if n % 3 == 0 then        return n == 3    end    if n % 5 == 0 then        return n == 5    end     local wheel = { 4, 2, 4, 2, 4, 6, 2, 6 }    local p = 7    while true do        for w = 1, #wheel do            if p * p > n then                return true            end            if n % p == 0 then                return false            end            p = p + wheel[w]        end    endend function digit_sum(n)    local sum = 0    while n > 0 do        sum = sum + n % 10        n = math.floor(n / 10)    end    return sumend local limit1 = 10000local limit2 = 1000000000local last = 10local p = 0local n = 0local extra_primes = {} print("Extra primes under " .. limit1 .. ":")while true do    p = next_prime_digit_number(p)    if p >= limit2 then        break    end    if is_prime(digit_sum(p)) and is_prime(p) then        n = n + 1        if p < limit1 then            print(string.format("%2d: %d", n, p))        end        extra_primes[n % last] = p    endend print(string.format("\nLast %d extra primes under %d:", last, limit2))local i = last - 1while i >= 0 do    print(string.format("%d: %d", n - i, extra_primes[(n - i) % last]))    i = i - 1end
Output:
Extra primes under 10000:
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727

Last 10 extra primes under 1000000000:
9049: 777753773
9050: 777755753
9051: 777773333
9052: 777773753
9053: 777775373
9054: 777775553
9055: 777775577
9056: 777777227
9057: 777777577
9058: 777777773

            NORMAL MODE IS INTEGER            BOOLEAN PRIME            DIMENSION PRIME(7777)            VECTOR VALUES FMT = $I4*$            PRINT COMMENT $EXTRA PRIMES UP TO 10000$             THROUGH SET, FOR P=1, 1, P.G.7777SET         PRIME(P) = 1B             THROUGH SIEVE, FOR P=2, 1, P*P.G.7777            THROUGH SIEVE, FOR C=P*P, P, C.G.7777SIEVE       PRIME(C) = 0B             THROUGH X, FOR VALUES OF A = 0,2,3,5,7            THROUGH X, FOR VALUES OF B = 0,2,3,5,7            WHENEVER A.NE.0 .AND. B.E.0, TRANSFER TO X            THROUGH Y, FOR VALUES OF C = 0,2,3,5,7            WHENEVER B.NE.0 .AND. C.E.0, TRANSFER TO Y            THROUGH Z, FOR VALUES OF D = 2,3,5,7            NUM = A*1000 + B*100 + C*10 + D            SUM = A+B+C+DZ           WHENEVER PRIME(NUM) .AND. PRIME(SUM),          0       PRINT FORMAT FMT, NUMY           CONTINUEX           CONTINUE             END OF PROGRAM
Output:
EXTRA PRIMES UP TO 10000
2
3
5
7
23
223
227
337
353
373
557
577
733
757
773
2333
2357
2377
2557
2753
2777
3253
3257
3323
3527
3727
5233
5237
5273
5323
5527
7237
7253
7523
7723
7727

## Mathematica/Wolfram Language

Select[Range[10000], PrimeQ[#] && AllTrue[IntegerDigits[#], PrimeQ] &]
Output:
{2,3,5,7,23,37,53,73,223,227,233,257,277,337,353,373,523,557,577,727,733,757,773,2237,2273,2333,2357,2377,2557,2753,2777,3253,3257,3323,3373,3527,3533,3557,3727,3733,5227,5233,5237,5273,5323,5333,5527,5557,5573,5737,7237,7253,7333,7523,7537,7573,7577,7723,7727,7753,7757}

import sequtils, strutils const N = 10_000 func isPrime(n: Positive): bool =  if (n and 1) == 0: return n == 2  var m = 3  while m * m <= n:    if n mod m == 0: return false    inc m, 2  result = true var primeList: seq[0..N]var primeSet: set[0..N] for n in 2..N:  if n.isPrime:    primeList.add n    primeSet.incl n type Digit = 0..9 proc digits(n: Positive): seq[Digit] =  var n = n.int  while n != 0:    result.add n mod 10    n = n div 10 proc isExtraPrime(prime: Positive): bool =  var sum = 0  for digit in prime.digits:    if digit notin primeSet: return false    inc sum, digit  result = sum in primeSet let result = primeList.filterIt(it.isExtraPrime)echo "Found $1 extra primes less than$2:".format(result.len, N)for i, p in result:  stdout.write (p).align(4) stdout.write if (i + 1) mod 9 == 0: '\n' else: ' ' Output: Found 36 extra primes less than 10000: 2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727 ## Perl Library: ntheory use strict;use warnings;use feature 'say';use ntheory qw(is_prime vecsum todigits forprimes); mystr;forprimes {    is_prime(vecsum(todigits($_))) and /^[2357]+$/ and $str .= sprintf '%-5d',$_;} 1e4;say $str =~ s/.{1,80}\K /\n/gr; Output: 2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727  ## Phix Minor reworking of Numbers_with_prime_digits_whose_sum_is_13#Phix#iterative constant limit = 1_000_000_000, --constant limit = 10_000, lim = limit/10-1, dgts = {2,3,5,7} function extra_primes() sequence res = {}, q = {{0,0}} integer s, -- partial digit sum v -- corresponding value while length(q) do {s,v} = q[1] q = q[2..$]
for i=1 to length(dgts) do
integer d = dgts[i], {ns,nv} = {s+d,v*10+d}
if is_prime(ns) and is_prime(nv) then res &= nv end if
if nv<lim then q &= {{ns,nv}} end if
end for
end while
return res
end function

atom t0 = time()
printf(1,"Extra primes < %,d:\n",{limit})
sequence res = extra_primes()
integer ml = min(length(res),37)
printf(1,"[1..%d]: %s\n",{ml,ppf(res[1..ml],{pp_Indent,9,pp_Maxlen,94})})
if length(res)>ml then
printf(1,"[991..1000]: %v\n",{res[991..1000]})
integer l = length(res)
printf(1,"[%d..%d]: %v\n",{l-8,l,res[l-8..l]})
end if
?elapsed(time()-t0)

Output:
Extra primes < 1,000,000,000:
[1..37]: {2,3,5,7,23,223,227,337,353,373,557,577,733,757,773,2333,2357,2377,2557,2753,2777,
3253,3257,3323,3527,3727,5233,5237,5273,5323,5527,7237,7253,7523,7723,7727,22573}
[991..1000]: {25337353,25353227,25353373,25353577,25355227,25355333,25355377,25357333,25357357,25357757}
[9050..9058]: {777755753,777773333,777773753,777775373,777775553,777775577,777777227,777777577,777777773}
"1.9s"


with the smaller limit in place:

Extra primes < 10,000:
[1..36]: {2,3,5,7,23,223,227,337,353,373,557,577,733,757,773,2333,2357,2377,2557,2753,2777,
3253,3257,3323,3527,3727,5233,5237,5273,5323,5527,7237,7253,7523,7723,7727}
"0.1s"


## Python

from itertools import *from functools import reduce class Sieve(object):    """Sieve of Eratosthenes"""    def __init__(self):        self._primes = []        self._comps = {}        self._max = 2;     def isprime(self, n):        """check if number is prime"""        if n >= self._max: self._genprimes(n)        return n >= 2 and n in self._primes     def _genprimes(self, max):        while self._max <= max:            if self._max not in self._comps:                self._primes.append(self._max)                self._comps[self._max*self._max] = [self._max]            else:                for p in self._comps[self._max]:                    ps = self._comps.setdefault(self._max+p, [])                    ps.append(p)                del self._comps[self._max]            self._max += 1 def extra_primes():    """Successively generate all extra primes."""    d = [2,3,5,7]    s = Sieve()    for cand in chain.from_iterable(product(d, repeat=r) for r in count(1)):        num = reduce(lambda x, y: x*10+y, cand)        if s.isprime(num) and s.isprime(sum(cand)): yield num for n in takewhile(lambda n: n < 10000, extra_primes()):    print(n)
Output:
2
3
5
7
23
223
227
337
353
373
557
577
733
757
773
2333
2357
2377
2557
2753
2777
3253
3257
3323
3527
3727
5233
5237
5273
5323
5527
7237
7253
7523
7723
7727

## Raku

For the time being, (Doctor?), I'm going to assume that the task is really "Sequence of primes with every digit a prime and the sum of the digits a prime". Outputting my own take on a reasonable display of results, compact and easily doable but exercising it a bit.

my @ppp = lazy flat 2, 3, 5, 7, 23, grep { .is-prime && .comb.sum.is-prime },               flat (2..*).map: { flat ([X~] (2, 3, 5, 7) xx $_) X~ (3, 7) }; put 'Terms < 10,000: '.fmt('%34s'), @ppp[^(@ppp.first: * > 1e4, :k)];put '991st through 1000th: '.fmt('%34s'), @ppp[990 .. 999];put 'Crossing 10th order of magnitude: ', @ppp[9055..9060]; Output:  Terms < 10,000: 2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727 991st through 1000th: 25337353 25353227 25353373 25353577 25355227 25355333 25355377 25357333 25357357 25357757 Crossing 10th order of magnitude: 777777227 777777577 777777773 2222222377 2222222573 2222225273 ## REXX Some optimization was done for the generation of primes, way more than was needed for this task's limit. If the limit is negative, the list of primes found isn't shown, but the count of primes found is always shown. /*REXX pgm finds & shows all primes whose digits are prime and the digits sum to a prime*/parse arg hi . /*obtain optional argument from the CL.*/if hi=='' | hi=="," then hi= 10000 /*Not specified? Then use the default.*/list= hi>=0; hi= abs(hi) /*set a switch; use the absolute value*/call genP /*invoke subroutine to generate primes.*/xp= 1 /*number of extra primes found (so far)*/$= 2                                             /*a list that holds "extra" primes.    */      do j=3  by 2  for (hi-1)%2                 /*search for numbers in this range.    */      if verify(j, 2357) \== 0  then iterate     /*J  must be comprised of prime digits.*/      s= left(j, 1)                    do k=2  for length(j)-1      /*only need to sum #s with #digits ≥ 4 */                    s= s + substr(j, k, 1)       /*sum some middle decimal digits of  J.*/                    end   /*k*/      if \!.s  then iterate                      /*Is the sum not equal to prime?  Skip.*/      if j<=hP then do                           /*J may be small enough to see if prime*/                    if \!.j  then iterate        /*is  J  a prime?   No, then skip it.  */                    end                          /*                             _____   */               else do p=1  while @.p**2<=j      /*perform division up to the  √  J     */                    if j//@.p==0  then iterate j /*J divisible by a prime?  Then ¬ prime*/                    end   /*p*/      xp= xp + 1                                 /*bump the count of primes found so far*/      if list  then $=$ j                       /*maybe append extra prime ───► $list.*/ end /*j*/say commas(xp) ' primes found whose digits are prime and the digits sum to a prime' , "and which are less than " commas(hi)word(. ":", list + 1)if list then say$                              /*maybe display the list ───► terminal.*/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 ?/*──────────────────────────────────────────────────────────────────────────────────────*/iSqrt: procedure; parse arg x;  r= 0;   q= 1;                  do while q<=x;  q=q*4;  end         do while q>1;  q= q%4;  _= x-r-q;  r= r%2;  if _>=0  then do;  x= _; r= r+q;  end         end   /*while*/;       return r/*──────────────────────────────────────────────────────────────────────────────────────*/genP:        @.1=2;  @.2=3;  @.3=5;  @.4=7;  @.5=11;  @.6=13      !.=0;  !.2=1;  !.3=1;  !.5=1;  !.7=1;  !.11=1;  !.13=1      high= max(9 * digits(), iSqrt(hi) )        /*enough primes for sums & primality ÷ */                   #= 6;      sq.#= @.# ** 2     /*define # primes; define squared prime*/            do [email protected].#+4  by 2  while  #<=high     /*continue on with the next odd prime. */            parse var  j  ''  -1  _              /*obtain the last digit of the  J  var.*/            if    _==5  then iterate; if j// 3==0  then iterate  /*J ÷ by 5?  J ÷ by  3?*/            if j//7==0  then iterate; if j//11==0  then iterate  /*J ÷ by 7?  J ÷ by 11?*/                                                 /* [↓]  divide by the primes.   ___    */                  do k=6  to #  while sq.k<=j    /*divide  J  by other primes ≤ √ J     */                  if j//@.k == 0  then iterate j /*÷ by prev. prime?  ¬prime     ___    */                  end   /*k*/                    /* [↑]   only divide up to     √ J     */            #=#+1;   @.#= j;  sq.#= j*j;  !.j= 1 /*bump number of primes; assign prime#.*/            end         /*j*/      hP= @.#;                 return #          /*hP:  is the highest prime generated. */
output   when using the default input:

(Shown at three-quarter size.)

36  primes found whose digits are prime and the digits sum to a prime and which are less than  10,000:
2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727

output   when using the input:     -100000
89  primes found whose digits are prime and the digits sum to a prime and which are less than  100,000.

output   when using the input:     -1000000
222  primes found whose digits are prime and the digits sum to a prime and which are less than  1,000,000.

output   when using the input:     -10000000
718  primes found whose digits are prime and the digits sum to a prime and which are less than  10,000,000.

output   when using the input:     -100000000
2,498  primes found whose digits are prime and the digits sum to a prime and which are less than  100,000,000.

output   when using the input:     -1000000000
9,058  primes found whose digits are prime and the digits sum to a prime and which are less than  1,000,000,000.


## Ring

 load "stdlib.ring" limit = 10000num = 0for n = 1 to limit    x1 = prime1(n)    x2 = prime2(n)    x3 = isprime(n)    if x1 = 1 and x2 = 1 and x3       num = num + 1       see "The " + num + "th Extra Prime is: " + n + nl    oknext func prime1(x)     pstr = string(x)     len = len(pstr)     count = 0     for n = 1 to len          if isprime(number(pstr[n]))            count = count + 1         ok     next     if count = len         return 1     else        return 0     ok func prime2(x)     pstr = string(x)     len = len(pstr)     sum = 0     for n = 1 to len         sum = sum + number(pstr[n])     next     if isprime(sum)        return 1     else        return 0     ok

Output:

The 1th Extra Prime is: 2
The 2th Extra Prime is: 3
The 3th Extra Prime is: 5
The 4th Extra Prime is: 7
The 5th Extra Prime is: 23
The 6th Extra Prime is: 223
The 7th Extra Prime is: 227
The 8th Extra Prime is: 337
The 9th Extra Prime is: 353
The 10th Extra Prime is: 373
The 11th Extra Prime is: 557
The 12th Extra Prime is: 577
The 13th Extra Prime is: 733
The 14th Extra Prime is: 757
The 15th Extra Prime is: 773
The 16th Extra Prime is: 2333
The 17th Extra Prime is: 2357
The 18th Extra Prime is: 2377
The 19th Extra Prime is: 2557
The 20th Extra Prime is: 2753
The 21th Extra Prime is: 2777
The 22th Extra Prime is: 3253
The 23th Extra Prime is: 3257
The 24th Extra Prime is: 3323
The 25th Extra Prime is: 3527
The 26th Extra Prime is: 3727
The 27th Extra Prime is: 5233
The 28th Extra Prime is: 5237
The 29th Extra Prime is: 5273
The 30th Extra Prime is: 5323
The 31th Extra Prime is: 5527
The 32th Extra Prime is: 7237
The 33th Extra Prime is: 7253
The 34th Extra Prime is: 7523
The 35th Extra Prime is: 7723
The 36th Extra Prime is: 7727


## Ruby

Translation of: Java
def nextPrimeDigitNumber(n)    if n == 0 then        return 2    end    if n % 10 == 2 then        return n + 1    end    if n % 10 == 3 or n % 10 == 5 then        return n + 2    end    return 2 + nextPrimeDigitNumber((n / 10).floor) * 10end def isPrime(n)    if n < 2 then        return false    end    if n % 2 == 0 then        return n == 2    end    if n % 3 == 0 then        return n == 3    end    if n % 5 == 0 then        return n == 5    end     wheel = [4, 2, 4, 2, 4, 6, 2, 6]    p = 7    loop do        for w in wheel            if p * p > n then                return true            end            if n % p == 0 then                return false            end            p = p + w        end    endend def digitSum(n)    sum = 0    while n > 0        sum = sum + n % 10        n = (n / 10).floor    end    return sumend LIMIT = 10000p = 0n = 0 print "Extra primes under %d:\n" % [LIMIT]while p < LIMIT    p = nextPrimeDigitNumber(p)    if isPrime(p) and isPrime(digitSum(p)) then        n = n + 1        print "%2d: %d\n" % [n, p]    endendprint "\n"
Output:
Extra primes under 10000:
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727

## Rust

// [dependencies]// primal = "0.3" fn is_prime(n: u64) -> bool {    primal::is_prime(n)} fn next_prime_digit_number(n: u64) -> u64 {    if n == 0 {        return 2;    }    match n % 10 {        2 => n + 1,        3 | 5 => n + 2,        _ => 2 + next_prime_digit_number(n / 10) * 10,    }} fn digit_sum(mut n: u64) -> u64 {    let mut sum = 0;    while n > 0 {        sum += n % 10;        n /= 10;    }    return sum;} fn main() {    let limit1 = 10000;    let limit2 = 1000000000;    let last = 10;    let mut p = 0;    let mut n = 0;    let mut extra_primes = vec![0; last];    println!("Extra primes under {}:", limit1);    loop {        p = next_prime_digit_number(p);        if p >= limit2 {            break;        }        if is_prime(digit_sum(p)) && is_prime(p) {            n += 1;            if p < limit1 {                println!("{:2}: {}", n, p);            }            extra_primes[n % last] = p;        }    }    println!("\nLast {} extra primes under {}:", last, limit2);    let mut i = last;    while i > 0 {        i -= 1;        println!("{}: {}", n - i, extra_primes[(n - i) % last]);    }}
Output:
Extra primes under 10000:
1: 2
2: 3
3: 5
4: 7
5: 23
6: 223
7: 227
8: 337
9: 353
10: 373
11: 557
12: 577
13: 733
14: 757
15: 773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727

Last 10 extra primes under 1000000000:
9049: 777753773
9050: 777755753
9051: 777773333
9052: 777773753
9053: 777775373
9054: 777775553
9055: 777775577
9056: 777777227
9057: 777777577
9058: 777777773


## Sidef

Simple solution:

say 1e4.primes.grep { .digits.all { .is_prime } && .sumdigits.is_prime }
Output:
[2, 3, 5, 7, 23, 223, 227, 337, 353, 373, 557, 577, 733, 757, 773, 2333, 2357, 2377, 2557, 2753, 2777, 3253, 3257, 3323, 3527, 3727, 5233, 5237, 5273, 5323, 5527, 7237, 7253, 7523, 7723, 7727]


Generate such primes from digits (faster):

func extra_primes(upto, base = 10) {     upto = prev_prime(upto+1)     var list = []    var digits = @(^base)     var prime_digits = digits.grep { .is_prime }    var end_digits   = prime_digits.grep { .is_coprime(base) }     list << prime_digits.grep { !.is_coprime(base) }...     for k in (0 .. upto.ilog(base)) {        prime_digits.variations_with_repetition(k, {|*a|            next if ([end_digits[0], a...].digits2num(base) > upto)            end_digits.each {|d|                var n = [d, a...].digits2num(base)                list << n if (n.is_prime && n.sumdigits(base).is_prime)            }        })    }     list.sort} with (1e4) { |n|    say "Extra primes <= #{n.commify}:"    say extra_primes(n).join(' ')} with (1000000000) {|n|    say "\nLast 10 extra primes <= #{n.commify}:"    say extra_primes(n).last(10).join(' ')}
Output:
Extra primes <= 10,000:
2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727

Last 10 extra primes <= 1,000,000,000:
777753773 777755753 777773333 777773753 777775373 777775553 777775577 777777227 777777577 777777773


## Swift

import Foundation let wheel = [4,2,4,2,4,6,2,6] func isPrime(_ number: Int) -> Bool {    if number < 2 {        return false    }    if number % 2 == 0 {        return number == 2    }    if number % 3 == 0 {        return number == 3    }    if number % 5 == 0 {        return number == 5    }    var p = 7    while true {        for w in wheel {            if p * p > number {                return true            }            if number % p == 0 {                return false            }            p += w        }    }} func nextPrimeDigitNumber(_ number: Int) -> Int {    if number == 0 {        return 2    }    switch number % 10 {    case 2:        return number + 1    case 3, 5:        return number + 2    default:        return 2 + nextPrimeDigitNumber(number/10) * 10    }} func digitSum(_ num: Int) -> Int {    var sum = 0    var n = num    while n > 0 {        sum += n % 10        n /= 10    }    return sum} func pad(string: String, width: Int) -> String {    if string.count >= width {        return string    }    return String(repeating: " ", count: width - string.count) + string} func commatize(_ number: Int) -> String {    let n = NSNumber(value: number)    return NumberFormatter.localizedString(from: n, number: .decimal)} let limit1 = 10000let limit2 = 1000000000let last = 10var p = nextPrimeDigitNumber(0)var n = 0 print("Extra primes less than \(commatize(limit1)):")while p < limit1 {    if isPrime(digitSum(p)) && isPrime(p) {        n += 1        print(pad(string: commatize(p), width: 5),              terminator: n % 10 == 0 ? "\n" : " ")    }    p = nextPrimeDigitNumber(p)} print("\n\nLast \(last) extra primes less than \(commatize(limit2)):") var extraPrimes = Array(repeating: 0, count: last)while p < limit2 {    if isPrime(digitSum(p)) && isPrime(p) {        n += 1        extraPrimes[n % last] = p    }    p = nextPrimeDigitNumber(p)} for i in stride(from: last - 1, through: 0, by: -1) {    print("\(commatize(n - i)): \(commatize(extraPrimes[(n - i) % last]))")}
Output:
Extra primes less than 10,000:
2     3     5     7    23   223   227   337   353   373
557   577   733   757   773 2,333 2,357 2,377 2,557 2,753
2,777 3,253 3,257 3,323 3,527 3,727 5,233 5,237 5,273 5,323
5,527 7,237 7,253 7,523 7,723 7,727

Last 10 extra primes less than 1,000,000,000:
9,049: 777,753,773
9,050: 777,755,753
9,051: 777,773,333
9,052: 777,773,753
9,053: 777,775,373
9,054: 777,775,553
9,055: 777,775,577
9,056: 777,777,227
9,057: 777,777,577
9,058: 777,777,773


## Wren

Library: Wren-math
Library: Wren-fmt
import "/math" for Intimport "/fmt" for Fmt var digits = [2, 3, 5, 7] // the only digits which are primesvar digits2 = [3, 7]      // a prime > 5 can't end in 2 or 5var candidates = [[2, 2], [3, 3], [5, 5], [7, 7]]  // [number, sum of its digits] for (a in digits) {    for (b in digits2) candidates.add([10*a + b, a + b])} for (a in digits) {    for (b in digits) {       for (c in digits2) candidates.add([100*a + 10*b + c, a + b + c])    }} for (a in digits) {    for (b in digits) {        for (c in digits) {            for (d in digits2) candidates.add([1000*a + 100*b + 10*c + d, a + b + c + d])        }    }} System.print("The extra primes under 10,000 are:")var count = 0for (cand in candidates) {   if (Int.isPrime(cand[0]) && Int.isPrime(cand[1])) {      count = count + 1      Fmt.print("$2d:$4d", count, cand[0])   }}
Output:
The extra primes under 10,000 are:
1:    2
2:    3
3:    5
4:    7
5:   23
6:  223
7:  227
8:  337
9:  353
10:  373
11:  557
12:  577
13:  733
14:  757
15:  773
16: 2333
17: 2357
18: 2377
19: 2557
20: 2753
21: 2777
22: 3253
23: 3257
24: 3323
25: 3527
26: 3727
27: 5233
28: 5237
29: 5273
30: 5323
31: 5527
32: 7237
33: 7253
34: 7523
35: 7723
36: 7727


## XPL0

func    IsPrime(N);     \Return 'true' if N is a prime numberint     N, I;[if N <= 1 then return false;for I:= 2 to sqrt(N) do        if rem(N/I) = 0 then return false;return true;]; int T, T2, N, M, I, S, D, P;[T:= [0, 2, 3, 5, 7];           \prime digitsT2:= [1, 10, 100, 1000];        \10^Ifor N:= 1 to \$7FFF_FFFF do        [M:= N;  S:= 0;  P:= 0;        for I:= 0 to 3 do                [M:= M/5;                D:= T(rem(0));                S:= S+D;                P:= P + D*T2(I);                if M = 0 then I:= 3;                if D = 0 then [S:= 0;  I:=3];                ];        if P >= 7777 then exit;        if IsPrime(S) then                if IsPrime(P) then                        [IntOut(0, P);  CrLf(0)];        ];]
Output:
2
3
5
7
23
223
227
337
353
373
557
577
733
757
773
2333
2357
2377
2557
2753
2777
3253
3257
3323
3527
3727
5233
5237
5273
5323
5527
7237
7253
7523
7723
7727