Prime numbers that are the sum of consecutive triangular numbers.
3, 19, 31, 83, 109, 199, 251, 409, 571, 631, 683, 829, 1091, 1489, 1999, 2341, 2531, 2971, 3529, 4621, 4789, 5051, 7039, 7211, 7669, 8779, 9721, 10459, 10711, 11171, 13681, 14851, 15131, 16069, 16381, 16883, 17659, 18731, 20011, 20359, 21683, 23251, 24851
1
The only prime triangular number is 3, the first term. Other terms are the sum of 3 or 6 consecutive triangular numbers.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Wikipedia, Polygonal number
(Mma) nn = 400; f = 3; s = Table[k ((f - 2) k - (f - 4))/2, {k, nn}]; t = Table[Select[Plus @@@ Partition[s, n, 1], PrimeQ], {n, 6}]; mx = Min[Select[Max /@ Rest[t], # > 0 &]]; Select[Union[Flatten[t]], # <= mx &]
Cf. A163251 (squares), S000791-S000799.
nonn
T. D. Noe, Dec 11 2015