A175132 Integers k such that the k-th triangular number is the sum of 2 consecutive primes.
8, 12, 15, 20, 23, 24, 35, 43, 44, 59, 60, 71, 75, 80, 84, 92, 99, 104, 128, 140, 147, 148, 155, 159, 200, 204, 216, 231, 251, 264, 288, 295, 303, 332, 336, 339, 344, 363, 384, 395, 420, 439, 440, 451, 455, 463, 467, 468, 495, 528, 539, 543, 560, 587, 588, 608
Offset: 1
Examples
8 is a term: 8*(8+1)/2 = 36 = 17 + 19. 12 is a term: 12*(12+1)/2 = 78 = 37 + 41.
Programs
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Mathematica
With[{upto=10^5},(Sqrt[Select[8ListConvolve[{1,1},Prime[Range[upto]]]+1,IntegerQ[Sqrt[#]]&]]-1)/2] (* Paolo Xausa, Nov 05 2023 *) -
PARI
lista(nn) = {for (i=1, nn, vsp = 1 + 8 *(prime(i) + prime(i+1)); if (issquare(vsp), v = sqrtint(vsp) - 1; if (v % 2 ==0, print1(v/2, ", "););););} \\ Michel Marcus, Jun 02 2013