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%I #6 Dec 29 2023 11:51:15
%S 2,5,19,13,181,59,463,439,2699,281,2309,541,8191,2141,6091,3697,11321,
%T 1889,38303,7019,24697,8933,42089,11159,39901,21319,61507,21839,
%U 266221,17851,182467,37633,104281,102103,173249,40609,386279,32719,229553
%N Consider the array T(n, m) = m-th prime of the form n*i(i+1)/2 +/- 1. This sequence is the main diagonal.
%C T(n, m) is a prime which is n times some triangular number plus or minus 1.
%C Eventually all primes, p, appear since (p +/-1) times 1(1+1)/2 equals (p +/- 1).
%e 1 | 2, 5, 7, 11, 29, 37, 67, 79, 137, 191, 211, 277, 379, 631, 821, ...
%e 2 | 3, 5, 7, 11, 13, 19, 29, 31, 41, 43, 71, 73, 89, 109, 131, ...
%e 3 | 2, 17, 19, 29, 31, 83, 107, 109, 197, 199, 233, 359, 409, 569, 571, ...
%e 4 | 3, 5, 11, 13, 23, 41, 59, 61, 83, 113, 179, 181, 263, 311, 313, ...
%e 5 | 29, 31, 139, 179, 181, 331, 389, 599, 601, 1049, 1051, 1381, 1499, 1889, 2029, ...
%e 6 | 5, 7, 17, 19, 37, 59, 61, 89, 127, 167, 269, 271, 331, 397, 467, ...
%e 7 | 41, 43, 71, 197, 251, 461, 463, 547, 839, 953, 1471, 1931, 1933, 2099, 2647, ...
%e 8 | 7, 23, 47, 79, 167, 223, 359, 439, 727, 839, 1087, 1223, 1367, 1847, 2207, ...
%e 9 | 53, 89, 251, 593, 701, 1223, 1709, 1889, 2699, 4463, 4751, 5669, 7019, 8513,10151, ...
%e 10 | 11, 29, 31, 59, 61, 101, 149, 151, 211, 281, 359, 449, 659, 661, 911, ...
%e 11 | 67, 109, 307, 397, 727, 857, 859, 1319, 1321, 2089, 2309, 2311, 3037, 3299, 3301, ...
%t T[n_, m_] := Block[{c = 0, k = 1, s = {}, trnglr}, While[c < m + 1, trnglr = n*k(k + 1)/2; If[ PrimeQ[trnglr - 1], c++; AppendTo[s, trnglr - 1]]; If[PrimeQ[trnglr + 1], c++; AppendTo[s, trnglr + 1]]; k++; s = Union@s]; s[[m]] ]; Table[T[n, n], {n, 40}]
%Y Cf. A000217, A124110, A125766, A125767, A125768, A125769.
%K nonn
%O 1,1
%A _Jonathan Vos Post_ & _Robert G. Wilson v_, Dec 01 2006
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