ClearAll[FindPrimeTriangles, FindPrimeTrianglesHelper]
FindPrimeTriangles[max_] :=
 Module[{count = 0, firstsolution, primes, primeQ},
  primes = PrimeQ[Range[2 max]];
  primeQ[n_] := primes[[n]];
  ClearAll[FindPrimeTrianglesHelper];
  FindPrimeTrianglesHelper[start_, remainder_, mxx_] :=
   Module[{last, nexts, r, newstart, newremainder},
    If[Length[remainder] > 0,
     last = Last[start];
     Do[
      r = remainder[[ri]];
      If[primeQ[last + r],
       newstart = Append[start, r];
       newremainder = Delete[remainder, ri];
       FindPrimeTrianglesHelper[newstart, newremainder, mxx]
       ]
      ,
      {ri, Length[remainder]}
      ]
     ,
     If[primeQ[Last[start] + mxx],
      count++;
      If[count == 1,
       Print[Append[start, mxx]]
       ]
      ]
     ]
    ];
  FindPrimeTrianglesHelper[{1}, Range[2, max - 1], max];
  count
  ]
Table[FindPrimeTriangles[S],{S, 2, 20}]