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
Note that the first term is the sum of just one triangular number. All other numbers appear to be the sum of 3 or 6 triangular numbers.
T. D. Noe, Plot of 1000 terms
T. D. Noe, Table of 1000 terms
Vicente Felipe Izquierdo, Puzzle 810: Primes and the sum of consecutive triangulars
(Mma) nn = 200; tri = Table[n (n + 1)/2, {n, 0, nn}]; t = Table[s = 0; Table[s = s + tri[[i]]; {s, i - n + 1}, {i, n, nn}], {n, nn}]; t2 = Sort[Flatten[t, 1]]; mx = Plus @@ Take[tri, -2]; t3 = Select[t2, #[[1]] < mx && PrimeQ[#[[1]]] &]; Transpose[t3][[1]]
nonn
T. D. Noe, Nov 22 2015