# 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.

## 11l

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!

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:
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

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;

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 68

Based on the Algol W sample.

BEGIN # find extra primes - numbers whose digits are prime and whose         #
# digit sum is prime                                                   #
# the digits can only be 2, 3, 5, 7                                    #
# other than 1 digit numbers, the first three digits                     #
# can be 0, 2, 3, 5, 7 and the final digit can only be 3 and 7           #
# which means there are at most 5^3 * 2 = 250 possible numbers           #
# so we will use trial division for primality testing                    #
# returns TRUE if n is prime, FALSE otherwise - uses trial division      #
PROC is prime = ( INT n )BOOL:
IF   n < 3       THEN n = 2
ELIF n MOD 3 = 0 THEN n = 3
ELIF NOT ODD n   THEN FALSE
ELSE
BOOL is a prime := TRUE;
FOR f FROM 5 BY 2 WHILE f * f <= n AND is a prime DO
is a prime := n MOD f /= 0
OD;
is a prime
FI # is prime # ;
# first four numbers ) i.e.the 1 digit primes ) as a special case        #
print( ( "    2    3    5    7" ) );
INT count := 4;
# 2, 3 and 5 digit numberrs                                              #
INT d1 := 0;
TO 5 DO
INT d2 := 0;
TO 5 DO
IF ( d1 + d2 ) = 0 OR d2 /= 0 THEN
INT d3 := 2;
TO 4 DO
FOR d4 FROM 3 BY 4 TO 7 DO
INT sum = d1 + d2 + d3 + d4;
INT n   = ( ( ( ( ( d1 * 10 ) + d2 ) * 10 ) + d3 ) * 10 ) + d4;
IF is prime( sum ) AND is prime( n ) THEN
# found a prime whose prime digits sum to a prime #
print( ( " ", whole( n, -4 ) ) );
IF ( count +:= 1 ) MOD 12 = 0 THEN print( ( newline ) ) FI
FI
OD;
d3 +:= 1;
IF d3 = 4 OR d3 = 6 THEN d3 +:= 1 FI
OD
FI;
d2 +:= 1;
IF d2 = 1 OR d2 = 4 OR d2 = 6 THEN d2 +:= 1 FI
OD;
d1 +:= 1;
IF d1 = 1 OR d1 = 4 OR d1 = 6 THEN d1 +:= 1 FI
OD
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

## 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
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

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

## Arturo

extraPrime?: function [n]->
all? @[
prime? n
prime? sum digits n
every? digits n => prime?
]

extraPrimesBelow10K: select 1..10000 => extraPrime?

loop split.every: 9 extraPrimesBelow10K 'x ->
print map x 's -> pad to :string s 5
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

## AWK

# 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

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

#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

## Delphi

Works with: Delphi version 6.0

Uses Delphi string to examine to sum and test digits.

function IsPrime(N: integer): boolean;
{Optimised prime test - about 40% faster than the naive approach}
var I,Stop: integer;
begin
if (N = 2) or (N=3) then Result:=true
else if (n <= 1) or ((n mod 2) = 0) or ((n mod 3) = 0) then Result:= false
else
begin
I:=5;
Stop:=Trunc(sqrt(N));
Result:=False;
while I<=Stop do
begin
if ((N mod I) = 0) or ((N mod (i + 2)) = 0) then exit;
Inc(I,6);
end;
Result:=True;
end;
end;

function IsExtraPrime(N: integer): boolean;
{Check if 1) The number is prime}
{2) All the digits in the number is prime}
{3) The sum of all digits is prime}
var S: string;
var I,Sum,D: integer;
begin
Result:=False;
if not IsPrime(N) then exit;
Sum:=0;
S:=IntToStr(N);
for I:=1 to Length(S) do
begin
D:=byte(S[I])-$30; if not IsPrime(D) then exit; Sum:=Sum+D; end; Result:=IsPrime(Sum); end; procedure ShowExtraPrimes(Memo: TMemo); {Show all extra-primes less than 10,000} var I: integer; var Cnt: integer; var S: string; begin Cnt:=0; S:=''; for I:=1 to 10000 do if IsExtraPrime(I) then begin Inc(Cnt); S:=S+Format('%5d',[I]); if (Cnt mod 9)=0 then S:=S+#$0D#$0A; end; Memo.Lines.Add(S); 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 ## F# ### The Function 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 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.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 : 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 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 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 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 ## Haskell import Data.Char ( digitToInt ) isPrime :: Int -> Bool isPrime n |n < 2 = False |otherwise = null$ filter (\i -> mod n i == 0 ) [2 .. root]
where
root :: Int
root = floor $sqrt$ fromIntegral n

condition :: Int -> Bool
condition n = isPrime n && all isPrime digits && isPrime ( sum digits )
where
digits :: [Int]
digits = map digitToInt ( show n )

solution :: [Int]
solution = filter condition [1..9999]
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]

## J

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

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 $p before the inner function that references it. # Input: the maximum width # Output: a stream def extraprimes: [2,3,5,7] as$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
| ($maxlen | wide) | select( add | is_prime) | join("") | tonumber | select(is_prime) ; # The task: 4|extraprimes Output: 2 3 5 7 23 ... 7253 7523 7723 7727 ## Julia using Primes function extraprimes(maxlen) for i in 1:maxlen, combo in Iterators.product([[2, 3, 5, 7] for _ in 1:i]...) if isprime(sum(combo)) n = evalpoly(10, combo) isprime(n) && println(n) end end end extraprimes(4) Output: 2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727 ## Kotlin Translation of: Java private fun nextPrimeDigitNumber(n: Int): Int { return if (n == 0) { 2 } else when (n % 10) { 2 -> n + 1 3, 5 -> n + 2 else -> 2 + nextPrimeDigitNumber(n / 10) * 10 } } private fun isPrime(n: Int): Boolean { if (n < 2) { return false } if (n and 1 == 0) { return n == 2 } if (n % 3 == 0) { return n == 3 } if (n % 5 == 0) { return n == 5 } val wheel = intArrayOf(4, 2, 4, 2, 4, 6, 2, 6) var p = 7 while (true) { for (w in wheel) { if (p * p > n) { return true } if (n % p == 0) { return false } p += w } } } private fun digitSum(n: Int): Int { var nn = n var sum = 0 while (nn > 0) { sum += nn % 10 nn /= 10 } return sum } fun main() { val limit = 10000 var p = 0 var n = 0 println("Extra primes under$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)) * 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

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

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

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:
primeSet.incl n

type Digit = 0..9

proc digits(n: Positive): seq[Digit] =
var n = n.int
while n != 0:
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$2:".format(result.len, N)
for i, p in result:
stdout.write (p).align(4) stdout.write if (i + 1) mod 9 == 0: '\n' else: ' ' Output: Found 36 extra primes less than 10000: 2 3 5 7 23 223 227 337 353 373 557 577 733 757 773 2333 2357 2377 2557 2753 2777 3253 3257 3323 3527 3727 5233 5237 5273 5323 5527 7237 7253 7523 7723 7727 ## Pascal ### Free Pascal using simple circular buffer for last n solutions.With crossing 10th order of magnitude like in Raku. program SpecialPrimes; // modified smarandache {IFDEF FPC}{$MODE DELPHI}{$OPTIMIZATION ON,ALL}{$ENDIF} {$IFDEF WINDOWS}{$APPTYPE CONSOLE}{$ENDIF}
uses
sysutils;
const
Digits : array[0..3] of Uint32 = (2,3,5,7);

var
//circular buffer
Last64 : array[0..63] of Uint64;
cnt,Limit : NativeUint;
LastIdx: Int32;

procedure OutLast(i:Int32);
var
idx: Int32;
begin
idx := LastIdx-i;
If idx < Low(Last64) then
idx += High(Last64)+1;
For i := i downto 1 do
begin
write(Last64[idx]:12);
inc(idx);
if idx > High(Last64) then
idx := Low(Last64);
end;
writeln;
end;

function isSmlPrime64(n:UInt32):boolean;inline;
//n must be >=0 and <=180 = 20 times digit 9, uses 80x86 BIT TEST
begin
EXIT(n in [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,
79,83,89,97,101,103,107,109,113,127,131,137,139,149,151,
157,163,167,173,179])
end;

function isPrime(n:UInt64):boolean;
const
deltaWheel : array[0..7] of byte =( 2, 6, 4, 2, 4, 2, 4, 6);
var
p : NativeUint;
WheelIdx : Int32;
begin
if n < 180 then
EXIT(isSmlPrime64(n));

result := false;

if n mod 2 = 0 then EXIT;
if n mod 3 = 0 then EXIT;
if n mod 5 = 0 then EXIT;

p := 1;
WheelIdx := High(deltaWheel);
repeat
inc(p,deltaWheel[WheelIdx]);
if p*p > n then
BREAK;
if n mod p = 0 then
EXIT;
dec(WheelIdx);
IF WheelIdx< Low(deltaWheel) then
wheelIdx := High(deltaWheel);
until false;
result := true;
end;

procedure Check(n:NativeUint);

Begin
if isPrime(n) then
begin
Last64[LastIdx] := n;
inc(LastIdx);
If LastIdx>High(Last64) then
LastIdx := Low(Last64);
inc(cnt);
IF (n < 10000) then
Begin
write(n:5,',');
if cnt mod 10 = 0 then
writeln;
if cnt = 36 then
writeln;
end
else
IF n > Limit then
Begin
OutLast(7);
Limit *=10;
end;
end;
end;

var
i,j,pot10,DgtLimit,n,DgtCnt,v : NativeUint;
dgt,
dgtsum : Int32;
Begin
Limit := 100000;
cnt := 0;
LastIdx := 0;
//Creating the numbers not the best way but all upto 11 digits take 0.05s
//here only 9 digits
i := 0;
pot10 := 1;
DgtLimit := 1;
v := 4;
repeat
repeat
j := i;
DgtCnt := 0;
pot10 := 1;
n := 0;
dgtsum := 0;
repeat
dgt := Digits[j MOD 4];
dgtsum += dgt;
n += pot10*Dgt;
j := j DIV 4;
pot10 *=10;
inc(DgtCnt);
until DgtCnt = DgtLimit;
if isPrime(dgtsum) then   Check(n);
inc(i);
until i=v;
//one more digit
v *=4;
i :=0;
inc(DgtLimit);
until DgtLimit= 12;
inc(LastIdx);
OutLast(7);
writeln('count: ',cnt);
end.
@TIO.RUN:
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,
75577       75773       77377       77557       77573       77773      222337
772573      773273      773723      775237      775273      777277     2222333
7775737     7775753     7777337     7777537     7777573     7777753    22222223
77755523    77757257    77757523    77773277    77773723    77777327   222222227
777775373   777775553   777775577   777777227   777777577   777777773  2222222377
7777772773  7777773257  7777773277  7777775273  7777777237  7777777327 22222222223
77777757773 77777773537 77777773757 77777775553 77777777533 77777777573 {{77777272733 63 places before last}}
count: 107308
Real time: 46.241 s CPU share: 99.38 %
@home: Real time: 8.615 s maybe much faster div ( Ryzen 5600G 4.4 Ghz vs Xeon 2.3 Ghz)
count < 1E
1E5, E6, E7,  E8,  E9,  E10,   E11
89,222,718,2498,9058,32189,107308

## Perl

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

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 ## Quackery 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 #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 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 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 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 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 // [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 = 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 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 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