Sum of square and cube digits of an integer are primes

Sum of square and cube digits of an integer are 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.

Task

Find and show here all positive integers n less than 100 where:

• the sum of the digits of the square of n is prime; and
• the sum of the digits of the cube of n is also prime.

Example

16 satisfies the task descrption because 16 x 16 = 256 has a digit sum of 13 which is prime and 16 x 16 x 16 = 4096 has a digit sum of 19 which is also prime.

ALGOL 68

Library: ALGOL 68-primes

<lang algol68>BEGIN # find numbers where the digit sums of the square and cube are prtime #

   INT max number = 99; # maximum number to consider #
   PR read "primes.incl.a68" PR
   []BOOL prime = PRIMESIEVE ( INT d sum := 9; # calculate the largest possible digit sum #
                               INT n     := max number * max number * max number;
                               WHILE ( n OVERAB 10 ) > 0 DO
                                   d sum +:= 9
                               OD;
                               d sum
                             );
   # returns the sum of the digits of n #
   OP  DIGITSUM = ( INT n )INT:
       BEGIN
           INT v      := ABS n;
           INT result := v MOD 10;
           WHILE ( v OVERAB 10 ) > 0 DO
               result +:= v MOD 10
           OD;
           result
       END # DIGITSUM # ;
   FOR i TO max number DO
       INT i2 = i * i;
       IF prime[ DIGITSUM i2 ] THEN
           IF prime[ DIGITSUM ( i * i2 ) ] THEN
               print( ( " ", whole( i, 0 ) ) )
           FI
       FI
   OD

END</lang>

Output:

 16 17 25 28 34 37 47 52 64

APL

<lang apl>(⊢(/⍨)∧/∘((2=0+.=⍳|⊢)∘(+/⍎¨∘⍕)¨*∘2 3)¨)⍳100</lang>

Output:

16 17 25 28 34 37 47 52 64

AWK

<lang AWK>

1. syntax: GAWK -f SUM_OF_SQUARE_AND_CUBE_DIGITS_OF_AN_INTEGER_ARE_PRIMES.AWK
2. converted from FreeBASIC

BEGIN {

   start = 1
   stop = 99
   for (i=start; i<=stop; i++) {
     if (is_prime(digit_sum(i^3,10)) && is_prime(digit_sum(i^2,10))) {
       printf("%5d%1s",i,++count%10?"":"\n")
     }
   }
   printf("\nSum of square and cube digits are prime %d-%d: %d\n",start,stop,count)
   exit(0)

} function digit_sum(n,b, s) { # digital sum of n in base b

   while (n) {
     s += n % b
     n = int(n/b)
   }
   return(s)

} function is_prime(n, d) {

   d = 5
   if (n < 2) { return(0) }
   if (n % 2 == 0) { return(n == 2) }
   if (n % 3 == 0) { return(n == 3) }
   while (d*d <= n) {
     if (n % d == 0) { return(0) }
     d += 2
     if (n % d == 0) { return(0) }
     d += 4
   }
   return(1)

} </lang>

Output:

   16    17    25    28    34    37    47    52    64
Sum of square and cube digits are prime 1-99: 9

BQN

<lang bqn># 'Library' functions from BQNCrate Digits ← 10 {⌽𝕗|⌊∘÷⟜𝕗⍟(↕1+·⌊𝕗⋆⁼1⌈⊢)} Prime ← 2=·+´0=(1+↕)⊸|

(∧˝∘⍉∘((Prime +´∘Digits)¨⋆⌜⟜2‿3))⊸/↕100</lang>

Output:

⟨ 16 17 25 28 34 37 47 52 64 ⟩

C

<lang c>#include <stdio.h>

1. include <stdbool.h>

int digit_sum(int n) {

   int sum;
   for (sum = 0; n; n /= 10) sum += n % 10;
   return sum;

}

/* The numbers involved are tiny */ bool prime(int n) {

   if (n<4) return n>=2;
   for (int d=2; d*d <= n; d++)
       if (n%d == 0) return false;
   return true;

}

int main() {

   for (int i=1; i<100; i++)
       if (prime(digit_sum(i*i)) & prime(digit_sum(i*i*i)))
           printf("%d ", i);
   printf("\n");
   return 0;

}</lang>

Output:

16 17 25 28 34 37 47 52 64

CLU

<lang clu>digit_sum = proc (n: int) returns (int)

   sum: int := 0
   while n>0 do
       sum := sum + n // 10
       n := n / 10
   end
   return(sum)

end digit_sum

% The numbers tested for primality are very small, % so this simple test suffices. prime = proc (n: int) returns (bool)

   if n<2 then return(false) end
   d: int := 2
   while d*d <= n do
       if n//d=0 then return(false) end
       d := d+1
   end
   return(true)

end prime

accept = proc (n: int) returns (bool)

   return(prime(digit_sum(n**2)) cand prime(digit_sum(n**3)))

end accept

start_up = proc ()

   po: stream := stream$primary_output()
   for i: int in int$from_to(1, 99) do
       if accept(i) then
           stream$puts(po, int$unparse(i) || " ")
       end
   end

end start_up</lang>

Output:

16 17 25 28 34 37 47 52 64

COBOL

<lang cobol> IDENTIFICATION DIVISION.

      PROGRAM-ID. SQUARE-CUBE-DIGITS-PRIME.
      
      DATA DIVISION.
      WORKING-STORAGE SECTION.
      01 NUMBER-SEARCH-VARS.
         03 CAND             PIC 9(6).
         03 SQUARE           PIC 9(6).
         03 CUBE             PIC 9(6).
         
      01 SUM-DIGITS-VARS.
         03 SUM-NUM          PIC 9(6).
         03 DIGITS           PIC 9 OCCURS 6 TIMES INDEXED BY D
                             REDEFINES SUM-NUM.
         03 SUM              PIC 99.
      
      01 PRIME-TEST-VARS.
         03 DIVISOR          PIC 99.
         03 DIV-TEST         PIC 99V9999.
         03 FILLER           REDEFINES DIV-TEST.
            05 FILLER        PIC 99.
            05 FILLER        PIC 9999.
               88 DIVISIBLE  VALUE ZERO.
         03 PRIME-FLAG       PIC X.
            88 PRIME         VALUE '*'.
      
      01 OUT-FMT             PIC Z9.
      
      PROCEDURE DIVISION.
      BEGIN.
          PERFORM CHECK-NUMBER VARYING CAND FROM 1 BY 1
          UNTIL CAND IS EQUAL TO 100.
          STOP RUN.
      
      CHECK-NUMBER.
          MULTIPLY CAND BY CAND GIVING SQUARE.
          MULTIPLY CAND BY SQUARE GIVING CUBE.
          MOVE SQUARE TO SUM-NUM.
          PERFORM SUM-DIGITS.
          PERFORM PRIME-TEST.
          IF PRIME,
              MOVE CUBE TO SUM-NUM,
              PERFORM SUM-DIGITS,
              PERFORM PRIME-TEST,
              IF PRIME,
                  MOVE CAND TO OUT-FMT,
                  DISPLAY OUT-FMT.
          
      SUM-DIGITS.
          MOVE ZERO TO SUM.
          PERFORM SUM-DIGIT VARYING D FROM 1 BY 1
          UNTIL D IS GREATER THAN 6.
          
      SUM-DIGIT.
          ADD DIGITS(D) TO SUM.
      
      PRIME-TEST.
          MOVE '*' TO PRIME-FLAG.
          PERFORM CHECK-DIVISOR VARYING DIVISOR FROM 2 BY 1
          UNTIL NOT PRIME, OR DIVISOR IS EQUAL TO SUM.
      
      CHECK-DIVISOR.
          DIVIDE SUM BY DIVISOR GIVING DIV-TEST.
          IF DIVISIBLE, MOVE SPACE TO PRIME-FLAG.</lang>

Output:

16
17
25
28
34
37
47
52
64

F#

This task uses Extensible Prime Generator (F#) <lang fsharp> // Sum of square and cube digits of an integer are primes. Nigel Galloway: December 22nd., 2021 let rec fN g=function 0->g |n->fN(g+n%10)(n/10) [1..99]|>List.filter(fun g->isPrime(fN 0 (g*g)) && isPrime(fN 0 (g*g*g)))|>List.iter(printf "%d "); printfn "" </lang>

Output:

16 17 25 28 34 37 47 52 64

FOCAL

<lang focal>01.10 F I=1,100;D 2 01.20 Q

02.10 F P=2,3;S N=I^P;D 3;D 4;I (C)2.3 02.20 T %2,I,! 02.30 R

03.10 S S=0 03.20 S M=FITR(N/10) 03.30 S S=S+(N-M*10) 03.40 S N=M 03.50 I (-N)3.2

04.10 S C=0 04.20 I (1-S)4.3;S C=-1;R 04.30 I (2-S)4.4;S C=0;R 04.40 F D=2,FSQT(S)+1;D 5;I (C)4.5 04.50 R

05.10 S Z=S/D 05.20 I (FITR(Z)-Z)5.3;S C=-1 05.30 R</lang>

Output:

= 16
= 17
= 25
= 28
= 34
= 37
= 47
= 52
= 64

FreeBASIC

<lang freebasic> function digsum(byval n as uinteger, b as const uinteger) as uinteger

   'digital sum of n in base b
   dim as integer s
   while n
       s+=n mod b
       n\=b
   wend
   return s

end function

function isprime(n as const uinteger) as boolean

   if n<2 then return false
   if n<4 then return true
   if n mod 2 = 0 then return false
   dim as uinteger i = 3
   while i*i<=n
       if n mod i = 0 then return false
       i+=2
   wend
   return true

end function

for n as uinteger = 1 to 99

   if isprime(digsum(n^3,10)) andalso isprime(digsum(n^2,10)) then print n;"   ";

next n</lang>

Output:

16   17   25   28   34   37   47   52   64

Haskell

<lang haskell>import Data.Bifunctor (first) import Data.Numbers.Primes (isPrime)

p :: Int -> Bool p =

 ((&&) . primeDigitSum . (^ 2))
   <*> (primeDigitSum . (^ 3))

---

TEST -------------------------

main :: IO () main = print $ filter p [2 .. 99]

---

GENERIC ------------------------

primeDigitSum :: Int -> Bool primeDigitSum = isPrime . digitSum 10

digitSum :: Int -> Int -> Int digitSum base = go

 where
   go 0 = 0
   go n = uncurry (+) (first go $ quotRem n base)</lang>

Output:

[16,17,25,28,34,37,47,52,64]

J

<lang j>((1*./@p:[:+/@|:10#.^:_1^&2 3)"0#]) i.100</lang>

Output:

16 17 25 28 34 37 47 52 64

Julia

<lang julia>using Primes

is_sqcubsumprime(n) = isprime(sum(digits(n*n))) && isprime(sum(digits(n*n*n))) println(filter(is_sqcubsumprime, 1:100)) # [16, 17, 25, 28, 34, 37, 47, 52, 64] </lang>

MAD

<lang MAD> NORMAL MODE IS INTEGER

           BOOLEAN PRIME
           DIMENSION PRIME(100)
           PRIME(0)=0B
           PRIME(1)=0B
           THROUGH SET, FOR P=2, 1, P.G.100

SET PRIME(P)=1B

           THROUGH SIEVE, FOR P=2, 1, P*P.G.100
           THROUGH SIEVE, FOR C=P*P, P, C.G.100

SIEVE PRIME(C)=0B

           THROUGH CHECK, FOR I=1, 1, I.GE.100
           WHENEVER .NOT.PRIME(DIGSUM.(I*I)), TRANSFER TO CHECK
           WHENEVER .NOT.PRIME(DIGSUM.(I*I*I)), TRANSFER TO CHECK
           PRINT FORMAT FMT, I

CHECK CONTINUE

           INTERNAL FUNCTION(N)
           ENTRY TO DIGSUM.
           SUM=0
           NN=N

LOOP WHENEVER NN.G.0

               NXT=NN/10
               SUM=SUM+NN-NXT*10
               NN=NXT
               TRANSFER TO LOOP
           END OF CONDITIONAL
           FUNCTION RETURN SUM
           END OF FUNCTION
           
           VECTOR VALUES FMT = $I2*$
           END OF PROGRAM </lang>

Output:

16
17
25
28
34
37
47
52
64

Perl

Library: ntheory

<lang perl>#!/usr/bin/perl

use strict; # https://rosettacode.org/wiki/Sum_of_square_and_cube_digits_of_an_integer_are_primes use warnings; use ntheory qw( is_prime vecsum );

my @results = grep

 is_prime( vecsum( split //, $_ ** 2 ) ) &&
 is_prime( vecsum( split //, $_ ** 3 ) ), 1 .. 100;

print "@results\n";</lang>

Output:

16 17 25 28 34 37 47 52 64

Phix

with javascript_semantics
function ipsd(integer n) return is_prime(sum(sq_sub(sprintf("%d",n),'0'))) end function
function scdp(integer n) return ipsd(n*n) and ipsd(n*n*n) end function
pp(filter(tagset(99),scdp))

Output:

{16,17,25,28,34,37,47,52,64}

Raku

<lang perl6>say ^100 .grep: { .².comb.sum.is-prime && .³.comb.sum.is-prime }</lang>

Output:

(16 17 25 28 34 37 47 52 64)

Ring

<lang ring> load "stdlib.ring" see "working..." +nl

limit = 100

for n = 1 to limit

   sums = 0
   sumc = 0
   sps = string(pow(n,2))
   spc = string(pow(n,3))
   for m = 1 to len(sps)
       sums = sums + sps[m]
   next
   for p = 1 to len(spc)
       sumc = sumc + spc[p]
   next
   if isprime(sums) and isprime(sumc)
      see "" + n + " "
   ok

next

see nl + "done..." + nl </lang>

Output:

working...
16 17 25 28 34 37 47 52 64 
done...

TinyBASIC

This can only go up to 31 because 32^3 is too big to fit in a signed 16-bit int. <lang tinybasic>REM N, the number to be tested REM D, the digital sum of its square or cube REM T, temporary variable REM Z, did D test as prime or not

   LET N = 1
10 LET T = N*N*N
   GOSUB 20
   GOSUB 30
   IF Z = 0 THEN GOTO 11
   LET T = N*N
   GOSUB 20
   GOSUB 30
   IF Z = 0 THEN GOTO 11
   PRINT N
11 IF N = 31 THEN END
   LET N = N + 1
   GOTO 10
20 LET D = 0
21 IF T = 0 THEN RETURN
   LET D = D + (T-(T/10)*10)
   LET T = T/10
   GOTO 21
30 LET Z = 0
   IF D < 2 THEN RETURN
   LET Z = 1
   IF D < 4 THEN RETURN
   LET Z = 0
   IF (D/2)*2 = D THEN RETURN
   LET T = 1
31 LET T = T + 2
   IF T*T>D THEN GOTO 32
   IF (D/T)*T=D THEN RETURN
   GOTO 31
32 LET Z = 1
   RETURN</lang>

Output:

16
17
25

28

Wren

Library: Wren-math

<lang ecmascript>import "./math" for Int

for (i in 1..99) {

   if (Int.isPrime(Int.digitSum(i*i)) && Int.isPrime(Int.digitSum(i*i*i))) System.write("%(i) ")

} System.print()</lang>

Output:

16 17 25 28 34 37 47 52 64 

XPL0

<lang XPL0>func IsPrime(N); \Return 'true' if N is prime int N, I; [if N <= 2 then return N = 2; if (N&1) = 0 then \even >2\ return false; for I:= 3 to sqrt(N) do

   [if rem(N/I) = 0 then return false;
   I:= I+1;
   ];

return true; ];

func SumDigits(N); \Return the sum of digits in N int N, Sum; [Sum:= 0; while N do

   [N:= N/10;
   Sum:= Sum + rem(0);
   ];

return Sum; ];

int N; [for N:= 0 to 100-1 do

   if IsPrime(SumDigits(N*N)) & IsPrime(SumDigits(N*N*N)) then
       [IntOut(0, N);  ChOut(0, ^ )];

]</lang>

Output:

16 17 25 28 34 37 47 52 64