Strange unique prime triplets

Strange unique prime triplets 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.

Integers   n,   m,   and   p   are   strange unique primes   if   n,   m,   and   p   are unique primes,   and the sum of     n + m + p     is also prime.

Task

•   Find all triplets of strange unique primes in which   n,   m,   and   p   are all less than   30.
•   (stretch goal)   Show the count (only) of all the triplets of strange unique primes in which   n,   m,   and   p   are all less than   1,000.

Python

Using sympy.primerange.

<lang python>from sympy import primerange

def strange_triplets(mx: int = 30) -> None:

   primes = list(primerange(0, mx))
   primes3 = set(primerange(0, 3 * mx))
   c = 0
   for i, n in enumerate(primes):
       for j, m in enumerate(primes[i + 1:], i + 1):
           for p in primes[j + 1:]:
               if n + m + p in primes3:
                   c += 1
                   print(f"{c:2}: {n:2}+{m:2}+{p:2} = {n + m + p}")

strange_triplets()</lang>

Output:

 1:  3+ 5+11 = 19
 2:  3+ 5+23 = 31
 3:  3+ 5+29 = 37
 4:  3+ 7+13 = 23
 5:  3+ 7+19 = 29
 6:  3+11+17 = 31
 7:  3+11+23 = 37
 8:  3+11+29 = 43
 9:  3+17+23 = 43
10:  5+ 7+11 = 23
11:  5+ 7+17 = 29
12:  5+ 7+19 = 31
13:  5+ 7+29 = 41
14:  5+11+13 = 29
15:  5+13+19 = 37
16:  5+13+23 = 41
17:  5+13+29 = 47
18:  5+17+19 = 41
19:  5+19+23 = 47
20:  5+19+29 = 53
21:  7+11+13 = 31
22:  7+11+19 = 37
23:  7+11+23 = 41
24:  7+11+29 = 47
25:  7+13+17 = 37
26:  7+13+23 = 43
27:  7+17+19 = 43
28:  7+17+23 = 47
29:  7+17+29 = 53
30:  7+23+29 = 59
31: 11+13+17 = 41
32: 11+13+19 = 43
33: 11+13+23 = 47
34: 11+13+29 = 53
35: 11+17+19 = 47
36: 11+19+23 = 53
37: 11+19+29 = 59
38: 13+17+23 = 53
39: 13+17+29 = 59
40: 13+19+29 = 61
41: 17+19+23 = 59
42: 19+23+29 = 71

REXX

<lang rexx></lang>

Ring

<lang ring> load "stdlib.ring"

num = 0 limit = 30

see "working..." + nl see "the strange primes are:" + nl

for n = 1 to limit

   for m = n+1 to limit
       for p = m+1 to limit
           sum = n+m+p
           if isprime(sum) and isprime(n) and isprime(m) and isprime(p)
              num = num + 1
              see "" + num + ": " + n + "+" + m + "+" + p + " = " + sum + nl
           ok
       next
   next

next

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

Output:

working...
the strange primes are:
1: 3+5+11 = 19
2: 3+5+23 = 31
3: 3+5+29 = 37
4: 3+7+13 = 23
5: 3+7+19 = 29
6: 3+11+17 = 31
7: 3+11+23 = 37
8: 3+11+29 = 43
9: 3+17+23 = 43
10: 5+7+11 = 23
11: 5+7+17 = 29
12: 5+7+19 = 31
13: 5+7+29 = 41
14: 5+11+13 = 29
15: 5+13+19 = 37
16: 5+13+23 = 41
17: 5+13+29 = 47
18: 5+17+19 = 41
19: 5+19+23 = 47
20: 5+19+29 = 53
21: 7+11+13 = 31
22: 7+11+19 = 37
23: 7+11+23 = 41
24: 7+11+29 = 47
25: 7+13+17 = 37
26: 7+13+23 = 43
27: 7+17+19 = 43
28: 7+17+23 = 47
29: 7+17+29 = 53
30: 7+23+29 = 59
31: 11+13+17 = 41
32: 11+13+19 = 43
33: 11+13+23 = 47
34: 11+13+29 = 53
35: 11+17+19 = 47
36: 11+19+23 = 53
37: 11+19+29 = 59
38: 13+17+23 = 53
39: 13+17+29 = 59
40: 13+19+29 = 61
41: 17+19+23 = 59
42: 19+23+29 = 71
done...