# Numbers whose binary and ternary digit sums are prime

Show positive integers   n   whose binary and ternary digits sum are prime,   where   n   <   200.

Numbers whose binary and ternary digit sums are prime 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.

## 11l

Translation of: Nim
```F is_prime(a)
I a == 2
R 1B
I a < 2 | a % 2 == 0
R 0B
L(i) (3 .. Int(sqrt(a))).step(2)
I a % i == 0
R 0B
R 1B

F digit_sum(=n, b)
V result = 0
L n != 0
result += n % b
n I/= b
R result

V count = 0
L(n) 2..199
I is_prime(digit_sum(n, 2)) &
is_prime(digit_sum(n, 3))
count++
print(‘#3’.format(n), end' I count % 16 == 0 {"\n"} E ‘ ’)
print()
print(‘Found ’count‘ numbers.’)```
Output:
```  5   6   7  10  11  12  13  17  18  19  21  25  28  31  33  35
36  37  41  47  49  55  59  61  65  67  69  73  79  82  84  87
91  93  97 103 107 109 115 117 121 127 129 131 133 137 143 145
151 155 157 162 167 171 173 179 181 185 191 193 199
Found 61 numbers.
```

## Action!

```INCLUDE "H6:SIEVE.ACT"

BYTE Func IsPrime(INT i BYTE base BYTE ARRAY primes)
BYTE sum,d

sum=0
WHILE i#0
DO
d=i MOD base
sum==+d
i==/base
OD
RETURN (primes(sum))

PROC Main()
DEFINE MAX="199"
BYTE ARRAY primes(MAX+1)
INT i,count=[0]

Put(125) PutE() ;clear the screen
Sieve(primes,MAX+1)
FOR i=1 TO MAX
DO
IF IsPrime(i,2,primes)=1 AND IsPrime(i,3,primes)=1 THEN
PrintI(i) Put(32)
count==+1
FI
OD
PrintF("%E%EThere are %I numbers",count)
RETURN```
Output:
```5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97
103 107 109 115 117 121 127 129 131 133 137 143 14 5 151 155 157 162 167 171 173 179 181 185 191 193 199

There are 61 numbers
```

## ALGOL 68

```BEGIN # find numbers whose digit sums in binary and ternary are prime #
# returns the digit sum of n in base b #
PRIO DIGITSUM = 9;
OP   DIGITSUM = ( INT n, b )INT:
BEGIN
INT d sum := 0;
INT v     := ABS n;
WHILE v > 0 DO
d sum +:= v MOD b;
v  OVERAB b
OD;
d sum
END # DIGITSUM # ;
INT max number = 200;
[]BOOL prime    = PRIMESIEVE 200;
INT    n count := 0;
FOR n TO UPB prime DO
INT d sum 2 = n DIGITSUM 2;
IF prime[ d sum 2 ] THEN
INT d sum 3 = n DIGITSUM 3;
IF prime[ d sum 3 ] THEN
# the base 2 and base 3 digit sums of n are both prime #
print( ( " ", whole( n, -3 ), IF prime[ n ] THEN "*" ELSE " " FI ) );
n count +:= 1;
IF n count MOD 14 = 0 THEN print( ( newline ) ) FI
FI
FI
OD;
print( ( newline ) );
print( ( "Found ", whole( n count, 0 ), " numbers whose binary and ternary digit sums are prime", newline ) );
print( ( "    those that are themselves prime are suffixed with a ""*""", newline ) )
END```
Output:
```   5*   6    7*  10   11*  12   13*  17*  18   19*  21   25   28   31*
33   35   36   37*  41*  47*  49   55   59*  61*  65   67*  69   73*
79*  82   84   87   91   93   97* 103* 107* 109* 115  117  121  127*
129  131* 133  137* 143  145  151* 155  157* 162  167* 171  173* 179*
181* 185  191* 193* 199*
Found 61 numbers whose binary and ternary digit sums are prime
those that are themselves prime are suffixed with a "*"
```

## ALGOL-M

```begin
integer function mod(a,b);
integer a,b;
mod := a-(a/b)*b;

integer function digitsum(n,base);
integer n,base;
digitsum := if n=0 then 0 else mod(n,base)+digitsum(n/base,base);

integer function isprime(n);
integer n;
begin
integer i;
isprime := 0;
if n < 2 then go to stop;
for i := 2 step 1 until n-1 do
begin
if mod(n,i) = 0 then go to stop;
end;
isprime := 1;
stop:
i := i;
end;

integer i,d2,d3,n;
n := 0;
for i := 0 step 1 until 199 do
begin
d2 := digitsum(i,2);
d3 := digitsum(i,3);
if isprime(d2) <> 0 and isprime(d3) <> 0 then
begin
if n/10 <> (n-1)/10 then write(i) else writeon(i);
n := n + 1;
end;
end;
end```
Output:
```     5     6     7    10    11    12    13    17    18    19
21    25    28    31    33    35    36    37    41    47
49    55    59    61    65    67    69    73    79    82
84    87    91    93    97   103   107   109   115   117
121   127   129   131   133   137   143   145   151   155
157   162   167   171   173   179   181   185   191   193
199```

## ALGOL W

```begin % find numbers whose binary and ternary digit sums are prime %
% returns the digit sum of n in base b %
integer procedure digitSum( integer value n, base ) ;
begin
integer v, dSum;
v    := abs n;
dSum := 0;
while v > 0 do begin
dSum := dSum + v rem base;
v    :=        v div base
end while_v_gt_0 ;
dSum
end digitSum ;
integer MAX_PRIME, MAX_NUMBER;
MAX_PRIME := 199;
begin
logical array prime( 1 :: MAX_PRIME );
integer       nCount;
% sieve the primes to MAX_PRIME %
prime( 1 ) := false; prime( 2 ) := true;
for i := 3 step 2 until MAX_PRIME do prime( i ) := true;
for i := 4 step 2 until MAX_PRIME do prime( i ) := false;
for i := 3 step 2 until truncate( sqrt( MAX_PRIME ) ) do begin
integer ii; ii := i + i;
if prime( i ) then for np := i * i step ii until MAX_PRIME do prime( np ) := false
end for_i ;
% find the numbers %
nCount := 0;
for i := 1 until MAX_PRIME do begin
if prime( digitSum( i, 2 ) ) and prime( digitSum( i, 3 ) ) then begin
% have another matching number %
writeon( i_w := 3, s_w := 0, " ", i );
nCount := nCount + 1;
if nCount rem 14 = 0 then write()
end if_have_a_suitable_number
end for_i ;
write( i_w := 1, s_w := 0, "Found ", nCount, " numbers with prime binary and ternary digit sums up to ", MAX_PRIME )
end
end.```
Output:
```   5   6   7  10  11  12  13  17  18  19  21  25  28  31
33  35  36  37  41  47  49  55  59  61  65  67  69  73
79  82  84  87  91  93  97 103 107 109 115 117 121 127
129 131 133 137 143 145 151 155 157 162 167 171 173 179
181 185 191 193 199
Found 61 numbers with prime binary and ternary digit sums up to 199
```

## APL

Works with: Dyalog APL
```(⊢(/⍨)(∧/((2=0+.=⍳|⊢)¨2 3(+/⊥⍣¯1)¨⊢))¨) ⍳200
```
Output:
```5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199```

## Arturo

```loop split.every: 10
select 1..199 'n [ and? prime? sum digits.base: 2 n
prime? sum digits.base: 3 n ] 'a ->
print map a => [pad to :string & 4]
```
Output:
```   5    6    7   10   11   12   13   17   18   19
21   25   28   31   33   35   36   37   41   47
49   55   59   61   65   67   69   73   79   82
84   87   91   93   97  103  107  109  115  117
121  127  129  131  133  137  143  145  151  155
157  162  167  171  173  179  181  185  191  193
199```

## AWK

```# syntax: GAWK -f NUMBERS_WHICH_BINARY_AND_TERNARY_DIGIT_SUM_ARE_PRIME.AWK
# converted from C
BEGIN {
start = 0
stop = 199
for (i=start; i<=stop; i++) {
if (is_prime(sum_digits(i,2)) && is_prime(sum_digits(i,3))) {
printf("%4d%1s",i,++count%10?"":"\n")
}
}
printf("\nBinary and ternary digit sums are both prime %d-%d: %d\n",start,stop,count)
exit(0)
}
function sum_digits(n,base,  sum) {
do {
sum += n % base
} while (n = int(n/base))
return(sum)
}
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:
```   5    6    7   10   11   12   13   17   18   19
21   25   28   31   33   35   36   37   41   47
49   55   59   61   65   67   69   73   79   82
84   87   91   93   97  103  107  109  115  117
121  127  129  131  133  137  143  145  151  155
157  162  167  171  173  179  181  185  191  193
199
Binary and ternary digit sums are both prime 0-199: 61
```

## BASIC

### Chipmunk Basic

Translation of: GW-BASIC
```100 rem Numbers which binary and ternary digit sum are prime
110 for n = 2 to 199
120   if is_prime(digit_sum(n,2)) <> 0 then
130     if is_prime(digit_sum(n,3)) <> 0 then print using "### ";n;
140   endif
150 next n
160 print
170 end
180 sub is_prime(p)
190   if p = 3 or p = 2 then
200     is_prime = 1
210   else
220     if p = 1 then
230       is_prime = 0
240     else
250       q = 1 : i = 1
260         i = i+1
270         if p mod i = 0 then q = 0
280       if i*i <= p and q <> 0 then 260
290       is_prime = q
300     endif
310   endif
320 end sub
330 sub digit_sum(n,bas)
340   p = 0 : xn = n
350   while xn <> 0
360     p = p+xn mod bas
370     xn = int(xn/bas)
380   wend
390   digit_sum = p
400 end sub
```
Output:
```  5   6   7  10  11  12  13  17  18  19  21  25  28  31  33  35  36  37  41  47  49  55  59  61  65  67  69  73  79  82  84  87  91  93  97 103 107 109 115 117 121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199
```

### FreeBASIC

```#include"isprime.bas"

function digsum( byval n as uinteger, b as const uinteger ) as uinteger
'finds the digit sum of n in base b
dim as uinteger sum = 0
while n
sum+=n mod b
n\=b
wend
return sum
end function

for n as uinteger = 1 to 200
if isprime(digsum(n,2)) and isprime(digsum(n,3)) then print n;"  ";
next n : print
```
Output:
`5  6  7  10  11  12  13  17  18  19  21  25  28  31  33  35  36  37  41  47  49  55  59  61  65  67  69  73  79  82  84  87  91  93  97  103  107  109  115  117  121  127  129  131  133  137  143  145  151  155  157  162  167  171  173  179  181  185  191  193  199`

### GW-BASIC

Works with: BASICA
```10  REM Numbers which binary and ternary digit sum are prime
20  FOR N = 2 TO 199
30   B = 2
40   GOSUB 200: GOSUB 120
50   IF Q = 0 THEN GOTO 90
60    B = 3
70    GOSUB 200: GOSUB 120
80    IF Q = 1 THEN PRINT USING "### ";N;
90  NEXT N
100 PRINT
110 END
120 REM tests if a number is prime
130 IF P = 3 OR P = 2 THEN Q = 1: RETURN
140 IF P = 1 THEN Q = 0: RETURN
150 Q = 1: I = 1
160  I = I + 1
170  IF P MOD I = 0 THEN Q = 0
180 IF I * I <= P AND Q <> 0 THEN GOTO 160
190 RETURN
200 REM finds the digit sum of N in base B, returns P
210 P = 0: XN = N
220 WHILE XN <> 0
230  P = P + XN MOD B
240  XN = XN \ B
250 WEND
260 RETURN
```
Output:
`  5   6   7  10  11  12  13  17  18  19  21  25  28  31  33  35  36  37  41  47  49  55  59  61  65  67  69  73  79  82  84  87  91  93  97 103 107 109 115 117 121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199`

Another solution

None of the digit sums are higher than 9, so the easiest thing to do is to hardcode which ones are prime.

Works with: BASICA
```10 REM Numbers which binary and ternary digit sum are prime
20 DEFINT I,J,K,P
30 DIM P(9): DATA 0,1,1,0,1,0,1,0,0
40 FOR I=1 TO 9: READ P(I): NEXT
50 FOR I=0 TO 199
60 J=0: K=I
70 IF K>0 THEN J=J+K MOD 2: K=K\2: GOTO 70 ELSE IF P(J)=0 THEN 100
80 J=0: K=I
90 IF K>0 THEN J=J+K MOD 3: K=K\3: GOTO 90 ELSE IF P(J) THEN PRINT I,
100 NEXT I
110 END
```
Output:
``` 5             6             7             10            11
12            13            17            18            19
21            25            28            31            33
35            36            37            41            47
49            55            59            61            65
67            69            73            79            82
84            87            91            93            97
103           107           109           115           117
121           127           129           131           133
137           143           145           151           155
157           162           167           171           173
179           181           185           191           193
199```

### Tiny BASIC

This isn't a very interesting problem. The most illustrative part of this solution is that it only uses four variables; several have multiple purposes. Efficiency is important when the language has only 26 variable names in total.

```    REM B      digital base input to sumdig, also output of primality routine
REM N      input to sumdig routine
REM P      input to primality routine, output of sumdig routine
REM T      temp variable in sumdig routine, loop var in prime routine

LET N = 1
20 LET N = N + 1
LET B = 2
GOSUB 200
GOSUB 100
IF B = 0 THEN GOTO 30
LET B = 3
GOSUB 200
GOSUB 100
IF B = 1 THEN PRINT N

30 IF N < 200 THEN GOTO 20
END

100 REM PRIMALITY BY TRIAL DIVISION
LET B = 0
IF P = 1 THEN RETURN
LET B = 1
IF P = 2 THEN RETURN
LET T = 2
110 IF (P/T)*T = P THEN LET B = 0
IF B = 0 THEN RETURN
LET T = T + 1
IF T*T <= P THEN GOTO 110
RETURN

200 REM digital sum of N in base B
LET T = N
LET P = 0
210 IF T = 0 THEN RETURN
LET P = P + T - (T/B)*B
LET T = T/B
GOTO 210```

## BCPL

```get "libhdr"

let digitsum(n, base) =
n=0 -> 0, n rem base + digitsum(n/base, base)

let isprime(n) = valof
\$(  if n<2 then resultis false
for i=2 to n-1 do
if n rem i = 0 then resultis false
resultis true
\$)

let accept(n) =
isprime(digitsum(n,2)) & isprime(digitsum(n,3))

let start() be
\$(  let c = 0
for i=0 to 199 do
if accept(i) do
\$(  writef("%I4",i)
c := c + 1
if c rem 10 = 0 then wrch('*N')
\$)
wrch('*N')
\$)```
Output:
```   5   6   7  10  11  12  13  17  18  19
21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82
84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155
157 162 167 171 173 179 181 185 191 193
199```

## C

```#include <stdio.h>
#include <stdint.h>

/* good enough for small numbers */
uint8_t prime(uint8_t n) {
uint8_t f;
if (n < 2) return 0;
for (f = 2; f < n; f++) {
if (n % f == 0) return 0;
}
return 1;
}

/* digit sum in given base */
uint8_t digit_sum(uint8_t n, uint8_t base) {
uint8_t s = 0;
do {s += n % base;} while (n /= base);
return s;
}

int main() {
uint8_t n, s = 0;
for (n = 0; n < 200; n++) {
if (prime(digit_sum(n,2)) && prime(digit_sum(n,3))) {
printf("%4d",n);
if (++s>=10) {
printf("\n");
s=0;
}
}
}
printf("\n");
return 0;
}
```
Output:
```   5   6   7  10  11  12  13  17  18  19
21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82
84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155
157 162 167 171 173 179 181 185 191 193
199```

## CLU

```prime = proc (n: int) returns (bool)
if n<2 then return(false) end
for i: int in int\$from_to(2, n-1) do
if n//i=0 then return(false) end
end
return(true)
end prime

digit_sum = proc (n, base: int) returns (int)
sum: int := 0
while n>0 do
sum := sum + n//base
n := n/base
end
return(sum)
end digit_sum

start_up = proc ()
po: stream := stream\$primary_output()
n: int := 0
for i: int in int\$from_to(2, 199) do
s2: int := digit_sum(i,2)
s3: int := digit_sum(i,3)
if prime(s2) cand prime(s3) then
stream\$putright(po, int\$unparse(i), 4)
n := n + 1
if n // 20 = 0 then stream\$putl(po, "") end
end
end
end start_up```
Output:
```   5   6   7  10  11  12  13  17  18  19  21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82  84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193
199```

## Cowgol

```include "cowgol.coh";

sub prime(n: uint8): (p: uint8) is
p := 0;
if n >= 2 then
var f: uint8 := 2;
while f < n loop
if n % f == 0 then
return;
end if;
f := f + 1;
end loop;
p := 1;
end if;
end sub;

sub digit_sum(n: uint8, base: uint8): (sum: uint8) is
sum := 0;
while n > 0 loop
sum := sum + n % base;
n := n / base;
end loop;
end sub;

var n: uint8 := 0;
while n < 200 loop;
if prime(digit_sum(n,2)) != 0 and prime(digit_sum(n,3)) != 0 then
print_i8(n);
print_nl();
end if;
n := n + 1;
end loop;```
Output:
```5
6
7
10
11
12
13
17
18
19
21
25
28
31
33
35
36
37
41
47
49
55
59
61
65
67
69
73
79
82
84
87
91
93
97
103
107
109
115
117
121
127
129
131
133
137
143
145
151
155
157
162
167
171
173
179
181
185
191
193
199```

## Delphi

Works with: Delphi version 6.0

```function IsPrime(N: int64): boolean;
{Fast, optimised prime test}
var I,Stop: int64;
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+0.0));
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 SumDigitsByBase(N,Base: integer): integer;
{Sum all digits of N in the specified B}
var I: integer;
begin
Result:=0;
repeat
begin
I:=N mod Base;
Result:=Result+I;
N:=N div Base;
end
until N = 0;
end;

function IsBinaryTernaryPrime(N: integer): boolean;
{Test if sums of binary and ternary digits is prime}
var Sum2,Sum3: integer;
begin
Result:=IsPrime(SumDigitsByBase(N,2)) and
IsPrime(SumDigitsByBase(N,3));
end;

procedure ShowBinaryTernaryPrimes(Memo: TMemo);
{Show the Binary-Ternary sum primes of first 200 values}
var I,Cnt: integer;
var S: string;
begin
S:=''; Cnt:=0;
for I:=0 to 200-1 do
if IsBinaryTernaryPrime(I) then
begin
Inc(Cnt);
S:=S+Format('%8D',[I]);
If (Cnt mod 5)=0 then S:=S+CRLF;
end;
end;
```
Output:
```       5       6       7      10      11
12      13      17      18      19
21      25      28      31      33
35      36      37      41      47
49      55      59      61      65
67      69      73      79      82
84      87      91      93      97
103     107     109     115     117
121     127     129     131     133
137     143     145     151     155
157     162     167     171     173
179     181     185     191     193
199
Count= 61
Elapsed Time: 3.322 ms.
```

## EasyLang

```fastfunc isprim num .
if num < 2
return 0
.
i = 2
while i <= sqrt num
if num mod i = 0
return 0
.
i += 1
.
return 1
.
func digsum n b .
while n > 0
sum += n mod b
n = n div b
.
return sum
.
for i = 1 to 199
if isprim digsum i 2 = 1 and isprim digsum i 3 = 1
write i & " "
.
.
```
Output:
```5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97 103 107 109 115 117 121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199
```

## F#

This task uses Extensible Prime Generator (F#)

```// binary and ternary digit sums are prime: Nigel Galloway. April 16th., 2021
let fN2,fN3=let rec fG n g=function l when l<n->l+g |l->fG n (g+l%n)(l/n) in (fG 2 0, fG 3 0)
{0..200}|>Seq.filter(fun n->isPrime(fN2 n) && isPrime(fN3 n))|>Seq.iter(printf "%d "); printfn ""
```
Output:
```5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97 103 107 109 115 117 121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199
Real: 00:00:00.005
```

## Factor

Works with: Factor version 0.99 2021-02-05
```USING: combinators combinators.short-circuit formatting io lists
lists.lazy math math.parser math.primes sequences ;

: dsum ( n base -- sum ) >base [ digit> ] map-sum ;
: dprime? ( n base -- ? ) dsum prime? ;
: 23prime? ( n -- ? ) { [ 2 dprime? ] [ 3 dprime? ] } 1&& ;
: l23primes ( -- list ) 1 lfrom [ 23prime? ] lfilter ;

: 23prime. ( n -- )
{
[ ]
[ >bin ]
[ 2 dsum ]
[ 3 >base ]
[ 3 dsum ]
} cleave
"%-8d %-9s %-6d %-7s %d\n" printf ;

"Base 10  Base 2    (sum)  Base 3  (sum)" print
l23primes [ 200 < ] lwhile [ 23prime. ] leach
```
Output:
```Base 10  Base 2    (sum)  Base 3  (sum)
5        101       2      12      3
6        110       2      20      2
7        111       3      21      3
10       1010      2      101     2
11       1011      3      102     3
12       1100      2      110     2
13       1101      3      111     3
17       10001     2      122     5
18       10010     2      200     2
19       10011     3      201     3
21       10101     3      210     3
25       11001     3      221     5
28       11100     3      1001    2
31       11111     5      1011    3
33       100001    2      1020    3
35       100011    3      1022    5
36       100100    2      1100    2
37       100101    3      1101    3
41       101001    3      1112    5
47       101111    5      1202    5
49       110001    3      1211    5
55       110111    5      2001    3
59       111011    5      2012    5
61       111101    5      2021    5
65       1000001   2      2102    5
67       1000011   3      2111    5
69       1000101   3      2120    5
73       1001001   3      2201    5
79       1001111   5      2221    7
82       1010010   3      10001   2
84       1010100   3      10010   2
87       1010111   5      10020   3
91       1011011   5      10101   3
93       1011101   5      10110   3
97       1100001   3      10121   5
103      1100111   5      10211   5
107      1101011   5      10222   7
109      1101101   5      11001   3
115      1110011   5      11021   5
117      1110101   5      11100   3
121      1111001   5      11111   5
127      1111111   7      11201   5
129      10000001  2      11210   5
131      10000011  3      11212   7
133      10000101  3      11221   7
137      10001001  3      12002   5
143      10001111  5      12022   7
145      10010001  3      12101   5
151      10010111  5      12121   7
155      10011011  5      12202   7
157      10011101  5      12211   7
162      10100010  3      20000   2
167      10100111  5      20012   5
171      10101011  5      20100   3
173      10101101  5      20102   5
179      10110011  5      20122   7
181      10110101  5      20201   5
185      10111001  5      20212   7
191      10111111  7      21002   5
193      11000001  3      21011   5
199      11000111  5      21101   5
```

## Fermat

```Function Digsum(n, b) =
digsum := 0;
while n>0 do
digsum := digsum + n|b;
n:=n\b;
od;
digsum.;

for p=1 to 200 do
if Isprime(Digsum(p,3)) and Isprime(Digsum(p,2)) then
!(p,'  ');
fi od;```
Output:
`5   6   7   10   11   12   13   17   18   19   21   25   28   31   33   35   36   37   41   47   49   55   59   61   65   67   69   73   79   82   84   87   91   93   97   103   107   109   115   117   121   127   129   131   133   137   143   145   151   155   157   162   167   171   173   179   181   185   191   193   199`

## FOCAL

```01.10 S P(2)=1;S P(3)=1;S P(5)=1;S P(7)=1
01.20 S V=10
01.30 F N=0,199;D 3
01.40 T !
01.50 Q

02.10 S A=0
02.20 S M=N
02.30 S T=FITR(M/B)
02.40 S A=A+M-T*B
02.50 S M=T
02.60 I (-M)2.3

03.10 S B=2;D 2;S X=A
03.20 S B=3;D 2;S Y=A
03.30 I (-P(X)*P(Y))3.4;R
03.40 T %4,N
03.50 S V=V-1
03.60 I (-V)3.7;T !;S V=10
03.70 R```
Output:
```=    5=    6=    7=   10=   11=   12=   13=   17=   18=   19
=   21=   25=   28=   31=   33=   35=   36=   37=   41=   47
=   49=   55=   59=   61=   65=   67=   69=   73=   79=   82
=   84=   87=   91=   93=   97=  103=  107=  109=  115=  117
=  121=  127=  129=  131=  133=  137=  143=  145=  151=  155
=  157=  162=  167=  171=  173=  179=  181=  185=  191=  193
=  199```

## Fōrmulæ

Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for storage and transfer purposes more than visualization and edition.

Programs in Fōrmulæ are created/edited online in its website.

In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.

Solution

## Go

Translation of: Wren
Library: Go-rcu
```package main

import (
"fmt"
"rcu"
)

func main() {
var numbers []int
for i := 2; i < 200; i++ {
bds := rcu.DigitSum(i, 2)
if rcu.IsPrime(bds) {
tds := rcu.DigitSum(i, 3)
if rcu.IsPrime(tds) {
numbers = append(numbers, i)
}
}
}
fmt.Println("Numbers < 200 whose binary and ternary digit sums are prime:")
for i, n := range numbers {
fmt.Printf("%4d", n)
if (i+1)%14 == 0 {
fmt.Println()
}
}
fmt.Printf("\n\n%d such numbers found\n", len(numbers))
}
```
Output:
```Numbers < 200 whose binary and ternary digit sums are prime:
5   6   7  10  11  12  13  17  18  19  21  25  28  31
33  35  36  37  41  47  49  55  59  61  65  67  69  73
79  82  84  87  91  93  97 103 107 109 115 117 121 127
129 131 133 137 143 145 151 155 157 162 167 171 173 179
181 185 191 193 199

61 such numbers found
```

```import Data.Bifunctor (first)
import Data.List.Split (chunksOf)
import Data.Numbers.Primes (isPrime)

--------- BINARY AND TERNARY DIGIT SUMS BOTH PRIME -------

digitSumsPrime :: Int -> [Int] -> Bool
digitSumsPrime n = all (isPrime . digitSum n)

digitSum :: Int -> Int -> Int
digitSum n base = go n
where
go 0 = 0
go n = uncurry (+) (first go \$ quotRem n base)

--------------------------- TEST -------------------------
main :: IO ()
main =
putStrLn \$
show (length xs)
<> " matches in [1..199]\n\n"
<> table xs
where
xs =
[1 .. 199]
>>= \x -> [show x | digitSumsPrime x [2, 3]]

------------------------- DISPLAY -----------------------

table :: [String] -> String
table xs =
let w = length (last xs)
in unlines \$
unwords
<\$> chunksOf
10
(justifyRight w ' ' <\$> xs)

justifyRight :: Int -> Char -> String -> String
justifyRight n c = (drop . length) <*> (replicate n c <>)
```
```61 matches in [1..199]

5   6   7  10  11  12  13  17  18  19
21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82
84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155
157 162 167 171 173 179 181 185 191 193
199```

## J

```((1*./@p:2 3+/@(#.^:_1)"0])"0#]) i.200
```
Output:
`5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97 103 107 109 115 117 121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199`

## jq

Works with: jq

Also works with gojq, the Go implementation of jq, and with fq

```def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l)[:\$l] + .;

def sum(s): reduce s as \$_ (0; . + \$_);

def is_prime:
. as \$n
| if (\$n < 2)         then false
elif (\$n % 2 == 0)  then \$n == 2
elif (\$n % 3 == 0)  then \$n == 3
elif (\$n % 5 == 0)  then \$n == 5
elif (\$n % 7 == 0)  then \$n == 7
elif (\$n % 11 == 0) then \$n == 11
elif (\$n % 13 == 0) then \$n == 13
elif (\$n % 17 == 0) then \$n == 17
elif (\$n % 19 == 0) then \$n == 19
else 23
| until( (. * .) > \$n or (\$n % . == 0); .+2)
| . * . > \$n
end;

def digitSum(\$base):
def stream:
recurse(if . > 0 then ./\$base|floor else empty end) | . % \$base ;
sum(stream);

def digit_sums_are_prime(\$n):
[range(2;\$n)
| digitSum(2) as \$bds
| select(\$bds|is_prime)
| digitSum(3) as \$tds
| select(\$tds|is_prime) ];

"Numbers < \(\$n) whose binary and ternary digit sums are prime:",
(digit_sums_are_prime(\$n)
| length as \$length
| (_nwise(14) | map(lpad(4)) | join(" ")),
"\nFound \(\$length) such numbers." );

Output:
```Numbers < 200 whose binary and ternary digit sums are prime:
5    6    7   10   11   12   13   17   18   19   21   25   28   31
33   35   36   37   41   47   49   55   59   61   65   67   69   73
79   82   84   87   91   93   97  103  107  109  115  117  121  127
129  131  133  137  143  145  151  155  157  162  167  171  173  179
181  185  191  193  199

Found 61 such numbers.
```

## Julia

```using Primes

btsumsareprime(n) = isprime(sum(digits(n, base=2))) && isprime(sum(digits(n, base=3)))

foreach(p -> print(rpad(p[2], 4), p[1] % 20 == 0 ? "\n" : ""), enumerate(filter(btsumsareprime, 1:199)))
```
Output:
```5   6   7   10  11  12  13  17  18  19  21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82  84  87  91  93  97  103 107 109 115 117
121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193
199
```

```            NORMAL MODE IS INTEGER

INTERNAL FUNCTION(P)
ENTRY TO PRIME.
WHENEVER P.L.2, FUNCTION RETURN 0B
THROUGH TEST, FOR DV=2, 1, DV.G.SQRT.(P)
TEST        WHENEVER P-P/DV*DV.E.0, FUNCTION RETURN 0B
FUNCTION RETURN 1B
END OF FUNCTION

INTERNAL FUNCTION(N,BASE)
ENTRY TO DGTSUM.
SUM = 0
DN = N
DIGIT       NX = DN/BASE
SUM = SUM + DN-NX*BASE
DN = NX
WHENEVER DN.G.0, TRANSFER TO DIGIT
FUNCTION RETURN SUM
END OF FUNCTION

THROUGH NBR, FOR I=0, 1, I.GE.200
WHENEVER PRIME.(DGTSUM.(I,2)) .AND. PRIME.(DGTSUM.(I,3))
PRINT FORMAT FMT, I
END OF CONDITIONAL
NBR         CONTINUE

VECTOR VALUES FMT = \$I3*\$
END OF PROGRAM```
Output:
```  5
6
7
10
11
12
13
17
18
19
21
25
28
31
33
35
36
37
41
47
49
55
59
61
65
67
69
73
79
82
84
87
91
93
97
103
107
109
115
117
121
127
129
131
133
137
143
145
151
155
157
162
167
171
173
179
181
185
191
193
199```

## Mathematica / Wolfram Language

```Partition[
Select[
Range@
200, (PrimeQ[Total@IntegerDigits[#, 2]] &&
PrimeQ[Total@IntegerDigits[#, 3]]) &], UpTo[8]] // TableForm
```
Output:
```
5	6	7	10	11	12	13	17
18	19	21	25	28	31	33	35
36	37	41	47	49	55	59	61
65	67	69	73	79	82	84	87
91	93	97	103	107	109	115	117
121	127	129	131	133	137	143	145
151	155	157	162	167	171	173	179
181	185	191	193	199

```

## Nim

```import strutils

func isPrime(n: Positive): bool =
if n == 1: return false
if n mod 2 == 0: return n == 2
if n mod 3 == 0: return n == 3
var d = 5
while d * d <= n:
if n mod d == 0: return false
inc d, 2
if n mod d == 0: return false
inc d, 4
return true

func digitSum(n, b: Natural): int =
var n = n
while n != 0:
result += n mod b
n = n div b

var count = 0
for n in 2..<200:
if digitSum(n, 2).isPrime and digitSum(n, 3).isPrime:
inc count
stdout.write (\$n).align(3), if count mod 16 == 0: '\n' else: ' '
echo()
echo "Found ", count, " numbers."
```
Output:
```  5   6   7  10  11  12  13  17  18  19  21  25  28  31  33  35
36  37  41  47  49  55  59  61  65  67  69  73  79  82  84  87
91  93  97 103 107 109 115 117 121 127 129 131 133 137 143 145
151 155 157 162 167 171 173 179 181 185 191 193 199
Found 61 numbers.```

## Perl

Library: ntheory
```use strict;
use warnings;
use feature 'say';
use List::Util 'sum';
use ntheory <is_prime todigitstring>;

sub test_digits { 0 != is_prime sum split '', todigitstring(shift, shift) }

my @p;
test_digits(\$_,2) and test_digits(\$_,3) and push @p, \$_ for 1..199;
say my \$result = @p . " matching numbers:\n" .  (sprintf "@{['%4d' x @p]}", @p) =~ s/(.{40})/\$1\n/gr;
```
Output:
```61 matching numbers:
5   6   7  10  11  12  13  17  18  19
21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82
84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155
157 162 167 171 173 179 181 185 191 193
199```

## Phix

```function to_base(atom n, integer base)
string result = ""
while true do
result &= remainder(n,base)
n = floor(n/base)
if n=0 then exit end if
end while
return result
end function

function prime23(integer n)
return is_prime(sum(to_base(n,2)))
and is_prime(sum(to_base(n,3)))
end function

sequence res = filter(tagset(199),prime23)
printf(1,"%d numbers found: %V\n",{length(res),shorten(res,"",5)})
```
Output:
```61 numbers found: {5,6,7,10,11,"...",181,185,191,193,199}
```

## PL/M

```100H:
/* CP/M CALLS */
BDOS: PROCEDURE (FN, ARG); DECLARE FN BYTE, ARG ADDRESS; GO TO 5; END BDOS;
EXIT: PROCEDURE; CALL BDOS(0,0); END EXIT;
PRINT: PROCEDURE (S); DECLARE S ADDRESS; CALL BDOS(9,S); END PRINT;

/* PRINT NUMBER */
PRINT\$NUMBER: PROCEDURE (N);
DECLARE S (8) BYTE INITIAL ('.....',13,10,'\$');
DECLARE (N, P) ADDRESS, C BASED P BYTE;
P = .S(5);
DIGIT:
P = P - 1;
C = N MOD 10 + '0';
N = N / 10;
IF N > 0 THEN GO TO DIGIT;
CALL PRINT(P);
END PRINT\$NUMBER;

/* SIMPLE PRIMALITY TEST */
PRIME: PROCEDURE (N) BYTE;
DECLARE (N, I) BYTE;
IF N < 2 THEN RETURN 0;
DO I=2 TO N-1;
IF N MOD I = 0 THEN RETURN 0;
END;
RETURN 1;
END PRIME;

/* SUM OF DIGITS */
DIGIT\$SUM: PROCEDURE (N, BASE) BYTE;
DECLARE (N, BASE, SUM) BYTE;
SUM = 0;
DO WHILE N > 0;
SUM = SUM + N MOD BASE;
N = N / BASE;
END;
RETURN SUM;
END DIGIT\$SUM;

/* TEST NUMBERS 0 .. 199 */
DECLARE I BYTE;
DO I=0 TO 199;
IF PRIME(DIGIT\$SUM(I,2)) AND PRIME(DIGIT\$SUM(I,3)) THEN
CALL PRINT\$NUMBER(I);
END;

CALL EXIT;
EOF```
Output:
```5
6
7
10
11
12
13
17
18
19
21
25
28
31
33
35
36
37
41
47
49
55
59
61
65
67
69
73
79
82
84
87
91
93
97
103
107
109
115
117
121
127
129
131
133
137
143
145
151
155
157
162
167
171
173
179
181
185
191
193
199```

## Plain English

```To run:
Start up.
Loop.
If a counter is past 200, break.
If the counter has prime digit sums in binary and ternary, write the counter then " " on the console without advancing.
Repeat.
Wait for the escape key.
Shut down.

A sum is a number.

A base is a number.

To find a digit sum of a number given a base:
Privatize the number.
Loop.
Divide the number by the base giving a quotient and a remainder.
Add the remainder to the digit sum.
Put the quotient into the number.
If the number is 0, exit.
Repeat.

To decide if a number has prime digit sums in binary and ternary:
Find a digit sum of the number given 2.
If the digit sum is not prime, say no.
Find another digit sum of the number given 3.
If the other digit sum is not prime, say no.
Say yes.```
Output:
```5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97 103 107 109 115 117 121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199
```

## Polyglot:PL/I and PL/M

Works with: 8080 PL/M Compiler

... under CP/M (or an emulator)

Should work with many PL/I implementations.
The PL/I include file "pg.inc" can be found on the Polyglot:PL/I and PL/M page. Note the use of text in column 81 onwards to hide the PL/I specifics from the PL/M compiler.

```/* FIND NUMBERS WHOSE DIGIT SUM SQUARED AND CUBED IS PRIME */
prime_digit_sums_100H: procedure options                                        (main);

/* PL/I DEFINITIONS                                                             */
%include 'pg.inc';
/* PL/M DEFINITIONS: CP/M BDOS SYSTEM CALL AND CONSOLE I/O ROUTINES, ETC. */    /*
DECLARE BINARY LITERALLY 'ADDRESS', CHARACTER LITERALLY 'BYTE';
DECLARE FIXED  LITERALLY ' ',       BIT       LITERALLY 'BYTE';
DECLARE STATIC LITERALLY ' ',       RETURNS   LITERALLY ' ';
DECLARE FALSE  LITERALLY '0',       TRUE      LITERALLY '1';
DECLARE HBOUND LITERALLY 'LAST',    SADDR     LITERALLY '.';
BDOSF: PROCEDURE( FN, ARG )BYTE;
DECLARE FN BYTE, ARG ADDRESS; GOTO 5;   END;
BDOS: PROCEDURE( FN, ARG ); DECLARE FN BYTE, ARG ADDRESS; GOTO 5;   END;
PRCHAR:   PROCEDURE( C );   DECLARE C BYTE;      CALL BDOS( 2, C ); END;
PRSTRING: PROCEDURE( S );   DECLARE S ADDRESS;   CALL BDOS( 9, S ); END;
PRNL:     PROCEDURE;        CALL PRCHAR( 0DH ); CALL PRCHAR( 0AH ); END;
PRNUMBER: PROCEDURE( N );
DECLARE V ADDRESS, N\$STR( 6 ) BYTE, W BYTE;
N\$STR( W := LAST( N\$STR ) ) = '\$';
N\$STR( W := W - 1 ) = '0' + ( ( V := N ) MOD 10 );
DO WHILE( ( V := V / 10 ) > 0 );
N\$STR( W := W - 1 ) = '0' + ( V MOD 10 );
END;
CALL BDOS( 9, .N\$STR( W ) );
END PRNUMBER;
RETURN( A MOD B );
END MODF;
/* END LANGUAGE DEFINITIONS */

DIGITSUM: PROCEDURE( N, BASE )RETURNS /* RETURNS THE DIGIT SUM OF N    */    (
FIXED BINARY          /* IN THE SPECIFIED BASE */    )
;
DECLARE ( N, BASE )  FIXED BINARY;
DECLARE ( SUM, V )   FIXED BINARY;
SUM = MODF( N, BASE );
V   = N / BASE;
DO WHILE( V > 0 );
SUM = SUM + MODF( V, BASE );
V   = V / BASE;
END;
RETURN( SUM );
END DIGITSUM ;

ISPRIME: PROCEDURE( N )RETURNS /* RETURNS TRUE  IF N IS PRIME, */            (
BIT             /* FALSE OTHERWISE      */            )
;
DECLARE N           FIXED BINARY;
DECLARE I           FIXED BINARY;
DECLARE RESULT      BIT;
IF      N < 2            THEN RESULT = FALSE;
ELSE IF N = 2            THEN RESULT = TRUE;
ELSE IF MODF( N, 2 ) = 0 THEN RESULT = FALSE;
ELSE DO;
RESULT = TRUE;
I      = 3;
DO WHILE( RESULT &                                                     /*
AND /* */ ( I * I ) <= N );
RESULT = MODF( N, I ) > 0;
I      = I + 2;
END;
END;
RETURN( RESULT );
END ISPRIME ;

DECLARE ( I, PCOUNT ) FIXED BINARY;

PCOUNT = 0;
DO I = 1 TO 199;
IF ISPRIME( DIGITSUM( I, 2 ) ) THEN DO;
IF ISPRIME( DIGITSUM( I, 3 ) ) THEN DO;
CALL PRCHAR( ' ' );
IF I <  10 THEN CALL PRCHAR( ' ' );
IF I < 100 THEN CALL PRCHAR( ' ' );
CALL PRNUMBER( I );
PCOUNT = PCOUNT + 1;
IF PCOUNT > 9 THEN DO;
PCOUNT = 0;
CALL PRNL;
END;
END;
END;
END;

EOF: end prime_digit_sums_100H;```
Output:
```   5   6   7  10  11  12  13  17  18  19
21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82
84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155
157 162 167 171 173 179 181 185 191 193
199
```

## Python

```'''Binary and Ternary digit sums both prime'''

# digitSumsPrime :: Int -> [Int] -> Bool
def digitSumsPrime(n):
'''True if the digits of n in each
given base have prime sums.
'''
def go(bases):
return all(
isPrime(digitSum(b)(n))
for b in bases
)
return go

# digitSum :: Int -> Int -> Int
def digitSum(base):
'''The sum of the digits of n in a given base.
'''
def go(n):
q, r = divmod(n, base)
return go(q) + r if n else 0
return go

# ------------------------- TEST -------------------------
# main :: IO ()
def main():
'''Matching integers in the range [1..199]'''
xs = [
str(n) for n in range(1, 200)
if digitSumsPrime(n)([2, 3])
]
print(f'{len(xs)} matches in [1..199]\n')
print(table(10)(xs))

# ----------------------- GENERIC ------------------------

# chunksOf :: Int -> [a] -> [[a]]
def chunksOf(n):
'''A series of lists of length n, subdividing the
contents of xs. Where the length of xs is not evenly
divible, the final list will be shorter than n.
'''
def go(xs):
return (
xs[i:n + i] for i in range(0, len(xs), n)
) if 0 < n else None
return go

# isPrime :: Int -> Bool
def isPrime(n):
'''True if n is prime.'''
if n in (2, 3):
return True
if 2 > n or 0 == n % 2:
return False
if 9 > n:
return True
if 0 == n % 3:
return False

def p(x):
return 0 == n % x or 0 == n % (2 + x)

return not any(map(p, range(5, 1 + int(n ** 0.5), 6)))

# table :: Int -> [String] -> String
def table(n):
'''A list of strings formatted as
rows of n (right justified) columns.
'''
def go(xs):
w = len(xs[-1])
return '\n'.join(
' '.join(row) for row in chunksOf(n)([
s.rjust(w, ' ') for s in xs
])
)
return go

# MAIN ---
if __name__ == '__main__':
main()
```
```61 matches in [1..199]

5   6   7  10  11  12  13  17  18  19
21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82
84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155
157 162 167 171 173 179 181 185 191 193
199```

## Quackery

`digitsum` is defined at Sum digits of an integer#Quackery.

`isprime` is defined at Primality by trial division#Quackery.

```  []
200 times
[ i^ 3 digitsum isprime while
i^ 2 digitsum isprime while
i^ join ]
echo```
Output:
`[ 5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97 103 107 109 115 117 121 127 129 131 133 137 143 145 151 155 157 162 167 171 173 179 181 185 191 193 199 ]`

## Raku

```say (^200).grep(-> \$n {all (2,3).map({\$n.base(\$_).comb.sum.is-prime}) }).batch(10)».fmt('%3d').join: "\n";
```
Output:
```  5   6   7  10  11  12  13  17  18  19
21  25  28  31  33  35  36  37  41  47
49  55  59  61  65  67  69  73  79  82
84  87  91  93  97 103 107 109 115 117
121 127 129 131 133 137 143 145 151 155
157 162 167 171 173 179 181 185 191 193
199```

## REXX

```/*REXX program finds and displays integers whose base 2 and base 3 digit sums are prime.*/
parse arg n cols .                               /*obtain optional argument from the CL.*/
if    n=='' |    n==","  then    n=  200         /*Not specified?  Then use the default.*/
if cols=='' | cols==","  then cols=   10         /* "      "         "   "   "     "    */
call genP                                        /*build array of semaphores for primes.*/
w= 10                                            /*width of a number in any column.     */
title= ' positive integers whose binary and ternary digit sums are prime, N  < ' commas(n)
if cols>0 then say ' index │'center(title, 1 + cols*(w+1)     )      /*maybe show title.*/
if cols>0 then say '───────┼'center(""   , 1 + cols*(w+1), '─')      /*maybe show  sep. */
found= 0;                   idx= 1               /*initialize # of finds and the index. */
\$=                                               /*a list of  numbers  found  (so far). */
do j=1  for n-1                               /*find #s whose B2 & B3 sums are prime.*/
b2= sumDig( tBase(j, 2) );   if \!.b2  then iterate   /*convert to base2, sum digits.*/      /* ◄■■■■■■■■ a filter. */
b3= sumDig( tBase(j, 3) );   if \!.b3  then iterate   /*   "     " base3   "    "    */      /* ◄■■■■■■■■ a filter. */
found= found + 1                              /*bump the number of  found integers.  */
if cols<1            then iterate             /*Only showing the summary?  Then skip.*/
\$= \$  right( commas(j), w)                    /*add a commatized integer ───► \$ list.*/
if found//cols\==0   then iterate             /*have we populated a line of output?  */
say center(idx, 7)'│'  substr(\$, 2);   \$=     /*display what we have so far  (cols). */
idx= idx + cols                               /*bump the  index  count for the output*/
end   /*j*/

if \$\==''  then say center(idx, 7)"│"  substr(\$, 2)  /*possible display residual output.*/
if cols>0  then say '───────┴'center(""   , 1 + cols*(w+1), '─')     /*show foot sep ?  */
say
say 'Found '       commas(found)      title                          /*show summary.    */
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 ?
sumDig: procedure; parse arg x 1 s 2;do j=2 for length(x)-1;s=s+substr(x,j,1);end;return s
/*──────────────────────────────────────────────────────────────────────────────────────*/
genP:   @= 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
!.=0;        do p=1  for words(@);   _= word(@, p);   !._= 1;   end;      return
/*──────────────────────────────────────────────────────────────────────────────────────*/
tBase:  procedure; parse arg x,toBase;      y=;       \$= 0123456789
do  while x>=toBase;   y= substr(\$, x//toBase+1, 1)y;   x= x % toBase
end   /*while*/
return substr(\$, x+1, 1)y
```
output   when using the default inputs:
``` index │                   positive integers whose binary and ternary digit sums are prime, N  <  200
───────┼───────────────────────────────────────────────────────────────────────────────────────────────────────────────
1   │          5          6          7         10         11         12         13         17         18         19
11   │         21         25         28         31         33         35         36         37         41         47
21   │         49         55         59         61         65         67         69         73         79         82
31   │         84         87         91         93         97        103        107        109        115        117
41   │        121        127        129        131        133        137        143        145        151        155
51   │        157        162        167        171        173        179        181        185        191        193
61   │        199
───────┴───────────────────────────────────────────────────────────────────────────────────────────────────────────────

Found  61  positive integers whose binary and ternary digit sums are prime, N  <  200
```

## Ring

```load "stdlib.ring"

see "working..." + nl
see "Numbers < 200 whose binary and ternary digit sums are prime:" + nl

decList = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]
baseList = ["0","1","2","3","4","5","6","7","8","9","A","B","C","D","E","F"]

num = 0
limit = 200

for n = 1 to limit
strBin = decimaltobase(n,2)
strTer = decimaltobase(n,3)
sumBin = 0
for m = 1 to len(strBin)
sumBin = sumBin + number(strBin[m])
next
sumTer = 0
for m = 1 to len(strTer)
sumTer = sumTer + number(strTer[m])
next
if isprime(sumBin) and isprime(sumTer)
num = num + 1
see "" + num + ". {" + n + "," + strBin + ":" + sumBin + "," + strTer + ":" + sumTer + "}" + nl
ok
next

see "Found " + num + " such numbers" + nl
see "done..." + nl

func decimaltobase(nr,base)
binList = []
binary = 0
remainder = 1
while(nr != 0)
remainder = nr % base
ind = find(decList,remainder)
rem = baseList[ind]
nr = floor(nr/base)
end
binlist = reverse(binList)
binList = list2str(binList)
binList = substr(binList,nl,"")
return binList```
Output:
```working...
Numbers < 200 whose binary and ternary digit sums are prime:
1. {5,101:2,12:3}
2. {6,110:2,20:2}
3. {7,111:3,21:3}
4. {10,1010:2,101:2}
5. {11,1011:3,102:3}
6. {12,1100:2,110:2}
7. {13,1101:3,111:3}
8. {17,10001:2,122:5}
9. {18,10010:2,200:2}
10. {19,10011:3,201:3}
11. {21,10101:3,210:3}
12. {25,11001:3,221:5}
13. {28,11100:3,1001:2}
14. {31,11111:5,1011:3}
15. {33,100001:2,1020:3}
16. {35,100011:3,1022:5}
17. {36,100100:2,1100:2}
18. {37,100101:3,1101:3}
19. {41,101001:3,1112:5}
20. {47,101111:5,1202:5}
21. {49,110001:3,1211:5}
22. {55,110111:5,2001:3}
23. {59,111011:5,2012:5}
24. {61,111101:5,2021:5}
25. {65,1000001:2,2102:5}
26. {67,1000011:3,2111:5}
27. {69,1000101:3,2120:5}
28. {73,1001001:3,2201:5}
29. {79,1001111:5,2221:7}
30. {82,1010010:3,10001:2}
31. {84,1010100:3,10010:2}
32. {87,1010111:5,10020:3}
33. {91,1011011:5,10101:3}
34. {93,1011101:5,10110:3}
35. {97,1100001:3,10121:5}
36. {103,1100111:5,10211:5}
37. {107,1101011:5,10222:7}
38. {109,1101101:5,11001:3}
39. {115,1110011:5,11021:5}
40. {117,1110101:5,11100:3}
41. {121,1111001:5,11111:5}
42. {127,1111111:7,11201:5}
43. {129,10000001:2,11210:5}
44. {131,10000011:3,11212:7}
45. {133,10000101:3,11221:7}
46. {137,10001001:3,12002:5}
47. {143,10001111:5,12022:7}
48. {145,10010001:3,12101:5}
49. {151,10010111:5,12121:7}
50. {155,10011011:5,12202:7}
51. {157,10011101:5,12211:7}
52. {162,10100010:3,20000:2}
53. {167,10100111:5,20012:5}
54. {171,10101011:5,20100:3}
55. {173,10101101:5,20102:5}
56. {179,10110011:5,20122:7}
57. {181,10110101:5,20201:5}
58. {185,10111001:5,20212:7}
59. {191,10111111:7,21002:5}
60. {193,11000001:3,21011:5}
61. {199,11000111:5,21101:5}
Found 61 such numbers
done...
```

## RPL

Works with: RPL version HP49-C
```« → base
« 0 SWAP
WHILE DUP NOT REPEAT
base MOD LASTARG / IP
UNROT + SWAP
END
DROP
» » 'BDSUM' STO

« { }
1 200 FOR j
IF j 3 BDSUM ISPRIME? THEN
IF j 2 BDSUM ISPRIME THEN
j +
END END
NEXT
```
Output:
```1: { 5 6 7 10 11 12 13 17 18 19 21 25 28 31 33 35 36 37 41 47 49 55 59 61 65 67 69 73 79 82 84 87 91 93 97 103 107 109 115 117 121 127 129 131 133 137 143 14 5 151 155 157 162 167 171 173 179 181 185 191 193 199 }
```

## Ruby

```require 'prime'

p (1..200).select{|n| [2, 3].all?{|base| n.digits(base).sum.prime?} }
```
Output:
```[5, 6, 7, 10, 11, 12, 13, 17, 18, 19, 21, 25, 28, 31, 33, 35, 36, 37, 41, 47, 49, 55, 59, 61, 65, 67, 69, 73, 79, 82, 84, 87, 91, 93, 97, 103, 107, 109, 115, 117, 121, 127, 129, 131, 133, 137, 143, 145, 151, 155, 157, 162, 167, 171, 173, 179, 181, 185, 191, 193, 199]
```

## Sidef

```1..^200 -> grep {|n| [2,3].all { n.sumdigits(_).is_prime } }
```
Output:
```[5, 6, 7, 10, 11, 12, 13, 17, 18, 19, 21, 25, 28, 31, 33, 35, 36, 37, 41, 47, 49, 55, 59, 61, 65, 67, 69, 73, 79, 82, 84, 87, 91, 93, 97, 103, 107, 109, 115, 117, 121, 127, 129, 131, 133, 137, 143, 145, 151, 155, 157, 162, 167, 171, 173, 179, 181, 185, 191, 193, 199]
```

## Wren

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

var numbers = []
for (i in 2..199) {
var bds = Int.digitSum(i, 2)
if (Int.isPrime(bds)) {
var tds = Int.digitSum(i, 3)
}
}
System.print("Numbers < 200 whose binary and ternary digit sums are prime:")
Fmt.tprint("\$4d", numbers, 14)
System.print("\nFound %(numbers.count) such numbers.")
```
Output:
```Numbers < 200 whose binary and ternary digit sums are prime:
5    6    7   10   11   12   13   17   18   19   21   25   28   31
33   35   36   37   41   47   49   55   59   61   65   67   69   73
79   82   84   87   91   93   97  103  107  109  115  117  121  127
129  131  133  137  143  145  151  155  157  162  167  171  173  179
181  185  191  193  199

Found 61 such numbers.
```

## 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;
];

func SumDigits(N, Base); \Return sum of digits in N for Base
int     N, Base, Sum;
[Sum:= 0;
repeat  N:= N/Base;
Sum:= Sum + rem(0);
until   N=0;
return Sum;
];

int Count, N;
[Count:= 0;
for N:= 0 to 200-1 do
if IsPrime(SumDigits(N,2)) & IsPrime(SumDigits(N,3)) then
[IntOut(0, N);
Count:= Count+1;
if rem(Count/10) = 0 then CrLf(0) else ChOut(0, 9\tab\);
];
CrLf(0);
IntOut(0, Count);
Text(0, " such numbers found below 200.
");
]```
Output:
```5       6       7       10      11      12      13      17      18      19
21      25      28      31      33      35      36      37      41      47
49      55      59      61      65      67      69      73      79      82
84      87      91      93      97      103     107     109     115     117
121     127     129     131     133     137     143     145     151     155
157     162     167     171     173     179     181     185     191     193
199
61 such numbers found below 200.
```