A212614 Least k > 1 such that the product tri(n) * tri(k) is triangular, or zero if no such k exists, where tri(k) is the k-th triangular number.

2, 5, 3, 6, 2, 4, 10, 0, 13, 7, 5, 4, 9, 3, 20, 208, 185, 14, 5, 2, 6, 14, 12, 115, 55, 37, 748, 11, 12, 1358, 90, 90, 6, 3, 21, 11, 26, 10, 33, 21, 265, 51, 61, 75, 96, 131, 201, 411, 0, 10, 7, 148, 113, 92, 4, 68, 364, 329, 50, 5083, 43, 329594, 38, 36, 2414

Offset: 1

Author

T. D. Noe, May 31 2012

Keywords

• nonn

Comments

That is, tri(k) = k(k+1)/2. It is provable that a(8) and a(49) are zero.

Other terms that are zero are given in sequence A001108. Note that a(71) = 2076978. In general, a Pell equation of the form x^2 = 1 + 2*n(n+1)*y*(y+1) must be solved to find a(n). - T. D. Noe, Jun 03 2012

Examples

			For n = 2, tri(n) = 3 and the first k is 5 because tri(5) = 15 and 3*15 = 45 is triangular.
		

Crossrefs

Cf. A188630 (triangular numbers that are tri(x) * tri(y) for some x,y > 1).

Cf. A212615 (similar sequence for pentagonal numbers).

Cf. A000217 (triangular numbers).

Programs

Mathematica

kMax = 10^6; TriangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; Table[t = n*(n+1)/2; k = 2; While[t2 = k*(k+1)/2; k < kMax && ! TriangularQ[t*t2], k++]; If[k == kMax, 0, k], {n, 65}]