Extra primes

From Rosetta Code
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.


Task

Show the extra primes under   10,000.


Reference

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


Related tasks



11l[edit]

V limit = 10'000

V is_prime = [0B] * 2 [+] [1B] * (limit - 1)
L(n) 0 .< Int(limit ^ 0.5 + 1.5)
   I is_prime[n]
      L(i) (n * n .< limit + 1).step(n)
         is_prime[i] = 0B

F is_extra_prime(n)
   I !:is_prime[n]
      R 0B
   V s = 0
   L(digit_char) String(n)
      V digit = Int(digit_char)
      I !:is_prime[digit]
         R 0B
      s += digit
   R Bool(:is_prime[s])

V i = 0
L(n) 0 .< limit
   I is_extra_prime(n)
      i++
      print(‘#4’.format(n), end' I i % 9 == 0 {"\n"} E ‘ ’)
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

Action![edit]

INCLUDE "H6:SIEVE.ACT"

BYTE Func IsExtraPrime(INT i BYTE ARRAY primes)
  BYTE sum,d

  IF primes(i)=0 THEN
    RETURN (0)
  FI

  sum=0
  WHILE i#0
  DO
    d=i MOD 10
    IF primes(d)=0 THEN
      RETURN (0)
    FI
    sum==+d
    i==/10
  OD
RETURN (primes(sum))

PROC Main()
  DEFINE MAX="9999"
  BYTE ARRAY primes(MAX+1)
  INT i,count=[0]

  Put(125) PutE()
  Sieve(primes,MAX+1)
  FOR i=2 TO MAX
  DO
    IF IsExtraPrime(i,primes) THEN
      PrintI(i) Put(32)
      count==+1
    FI
  OD
  PrintF("%E%EThere are %I extra primes",count)
RETURN
Output:

Screenshot from Atari 8-bit computer

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 523 7 5273 5323 5527 7237 7253 7523 7723 7727

There are 36 extra primes

Ada[edit]

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[edit]

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
    end
end.
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[edit]

extraPrimes{
    pd0 2 3 5 7
    ds↓⍉(dspd)/ds10(¯1)1↓⍳
    ds((/2(≤≥0=⊢)/)¨ds)/ds
    ns(ns)/ns10⊥⍉ds
    ss+/(ns)ds
    sieve~(1+⌈/ns,ss){
        r1()1
        /r:(r≠⍳-1)⍺∇1+2*r1
        (-1)/0
    }2
    (sieve[ns]sieve[ss])/ns
}
Output:
      extraPrimes 10000
2 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[edit]

# syntax: GAWK -f EXTRA_PRIMES.AWK
BEGIN {
    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[edit]

10 DEFINT A-Z: DIM S(7777),D(4): DATA 0,2,3,5,7
15 FOR I=0 TO 4: READ D(I): NEXT
20 FOR I=2 TO SQR(7777)
30 FOR J=I*I TO 7777 STEP I: S(J)=-1: NEXT
40 NEXT
50 FOR A=0 TO 4
60 FOR B=0 TO 4: IF A<>0 AND B=0 THEN 130
70 FOR C=0 TO 4: IF B<>0 AND C=0 THEN 120
80 FOR D=1 TO 4
90 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 NEXT
120 NEXT
130 NEXT
140 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[edit]

#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++[edit]

#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[edit]

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[edit]

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#[edit]

The Function[edit]

This task uses Permutations/Derangements#F.23

// Extra Primes. Nigel Galloway: January 9th., 2021
let 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

The tasks[edit]

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[edit]

Translation of: Wren
Works with: Factor version 0.99 2020-08-14
USING: formatting io kernel math math.functions math.primes
sequences 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:" print
candidates
[ 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[edit]

: 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:" cr
10000 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[edit]

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 true
end function

print "0002"  'special case
for 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 d2
next 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

Frink[edit]

select[primes[2,10000], {|n| isPrime[sum[integerDigits[n]]] and isSubset[toSet[integerDigits[n]], new set[2,3,5,7]]}]
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]

Go[edit]

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[edit]

exprimes =: (] #~ *./@(1&p:)@(+/ , ])@(10 #.^:_1 ])"0)@(i.&.(p:^:_1))
Output:
   exprimes 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

Java[edit]

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[edit]

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 stream
def 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[edit]

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
    end
end

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[edit]

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[edit]

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)) * 10
end

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
    end
end

function digit_sum(n)
    local sum = 0
    while n > 0 do
        sum = sum + n % 10
        n = math.floor(n / 10)
    end
    return sum
end

local limit1 = 10000
local limit2 = 1000000000
local last = 10
local p = 0
local n = 0
local 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
    end
end

print(string.format("\nLast %d extra primes under %d:", last, limit2))
local i = last - 1
while i >= 0 do
    print(string.format("%d: %d", n - i, extra_primes[(n - i) % last]))
    i = i - 1
end
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

MAD[edit]

            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.7777
SET         PRIME(P) = 1B
            
            THROUGH SIEVE, FOR P=2, 1, P*P.G.7777
            THROUGH SIEVE, FOR C=P*P, P, C.G.7777
SIEVE       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+D
Z           WHENEVER PRIME(NUM) .AND. PRIME(SUM),
          0       PRINT FORMAT FMT, NUM
Y           CONTINUE
X           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[edit]

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}

Nim[edit]

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[edit]

Library: ntheory
use strict;
use warnings;
use feature 'say';
use ntheory qw(is_prime vecsum todigits forprimes);

my $str;
forprimes {
    is_prime(vecsum(todigits($_))) 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[edit]

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[edit]

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

Quackery[edit]

eratosthenes and isprime are defines at Sieve of Eratosthenes#Quckery.

  [ [] swap
    [ 10 /mod
      rot join swap
      dup 0 = until ]
    drop ]              is digits ( n --> [ ) 

  [ 0 swap witheach + ] is sum    ( [ --> n )

  10000 eratosthenes
  []
  10000 times
    [ i^ isprime not if done
      true i^ digits
      tuck witheach
        [ isprime and
          dup not if conclude ]
      not iff drop done
      sum isprime
      if [ i^ join ] ]
  echo
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 ]


Racket[edit]

#lang racket

(require math/number-theory)

(define (extra-prime? p)
  (define (prime-sum-of-prime-digits? p (s 0))
    (if (zero? p)
        (prime? s)
        (let-values (((q r) (quotient/remainder p 10)))
          (case r
            ((2 3 5 7) (prime-sum-of-prime-digits? q (+ s r)))
            (else #f)))))
  (and (prime? p) (prime-sum-of-prime-digits? p)))

(displayln (filter extra-prime? (range 10000)))
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[edit]

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[edit]

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 j=@.#+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[edit]

load "stdlib.ring"

limit = 10000
num = 0
for 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
    ok
next

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[edit]

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) * 10
end

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
    end
end

def digitSum(n)
    sum = 0
    while n > 0
        sum = sum + n % 10
        n = (n / 10).floor
    end
    return sum
end

LIMIT = 10000
p = 0
n = 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]
    end
end
print "\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[edit]

// [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[edit]

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[edit]

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 = 10000
let limit2 = 1000000000
let last = 10
var 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[edit]

Library: Wren-math
Library: Wren-fmt
import "/math" for Int
import "/fmt" for Fmt

var digits = [2, 3, 5, 7] // the only digits which are primes
var digits2 = [3, 7]      // a prime > 5 can't end in 2 or 5
var 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 = 0
for (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[edit]

func    IsPrime(N);     \Return 'true' if N is a prime number
int     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 digits
T2:= [1, 10, 100, 1000];        \10^I
for 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