Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #15 Jun 07 2012 16:20:07
%S 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 748,11,12,1358,90,90,6,3,21,11,26,10,33,21,265,51,61,75,96,131,201,
%U 411,0,10,7,148,113,92,4,68,364,329,50,5083,43,329594,38,36,2414
%N 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 That is, tri(k) = k(k+1)/2. It is provable that a(8) and a(49) are zero.
%C 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 For n = 2, tri(n) = 3 and the first k is 5 because tri(5) = 15 and 3*15 = 45 is triangular.
%t 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 Cf. A188630 (triangular numbers that are tri(x) * tri(y) for some x,y > 1).
%Y Cf. A212615 (similar sequence for pentagonal numbers).
%Y Cf. A000217 (triangular numbers).
%K nonn
%O 1,1
%A _T. D. Noe_, May 31 2012