cp's OEIS Frontend

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.

%I A212614 #15 Jun 07 2012 16:20:07
%S A212614 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,
%T A212614 748,11,12,1358,90,90,6,3,21,11,26,10,33,21,265,51,61,75,96,131,201,
%U A212614 411,0,10,7,148,113,92,4,68,364,329,50,5083,43,329594,38,36,2414
%N 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.
%C A212614 That is, tri(k) = k(k+1)/2. It is provable that a(8) and a(49) are zero.
%C A212614 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
%e A212614 For n = 2, tri(n) = 3 and the first k is 5 because tri(5) = 15 and 3*15 = 45 is triangular.
%t A212614 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}]
%Y A212614 Cf. A188630 (triangular numbers that are tri(x) * tri(y) for some x,y > 1).
%Y A212614 Cf. A212615 (similar sequence for pentagonal numbers).
%Y A212614 Cf. A000217 (triangular numbers).
%K A212614 nonn
%O A212614 1,1
%A A212614 _T. D. Noe_, May 31 2012