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%I
%S 36,45,210,630,780,990,1540,2850,3570,4095,4851,8778,11781,15400,
%T 17955,19110,21528,25200,26565,26796,33930,37128,40755,43956,61425,
%U 61776,70125,79800,105570,113050,122265,145530,176715,189420,192510,246753,270480,303810,349866,437580,500500,526851
%N Triangular numbers that are the product of two triangular numbers greater than 1.
%C For squares, it is a simple matter to find squares that are the product of squares greater than 1. Is there a simple procedure for triangular numbers? That is, given n, is it easy to determine whether T(n) is the product of T(i) * T(j) for some i,j > 1?
%C Breiteig mentions this problem, but does not solve it. The problem can be extended to any polygonal number; for example, when is a pentagonal number the product of two pentagonal numbers? See A188660 and A188663 for the oblong and pentagonal cases.
%C Sequence A001571 gives the indices of triangular numbers that are 3 times another triangular number. For example, A001571(4) is 132; T(132) is 8778, which equals 3*T(76). Note that A061278 is the companion sequence, whose 4th term is 76. As with the oblong numbers covered by Breiteig, the triangular numbers in this sequence appear to satisfy linear recursions.
%H Trygve Breiteig, <a href="http://www.jstor.org/pss/2691083">When is the product of two oblong numbers another oblong?</a>, Math. Mag. 73 (2000), 120-129.
%H Donovan Johnson, <a href="/A188630/b188630.txt">Table of n, a(n) for n = 1..4082</a>
%e 210 = T(20) = 10 * 21 = T(4) * T(6).
%t TriangularQ[n_] := IntegerQ[Sqrt[1 + 8 n]]; TriIndex[n_] := Floor[(-1 + Sqrt[1 + 8*n])/2]; lim = 10^6; nMax = TriIndex[lim/3]; tri = Table[n (n + 1)/2, {n, 2, nMax}]; Union[Reap[Do[num = tri[[i]]*tri[[j]]; If[TriangularQ[num], Sow[num]], {i, TriIndex[Sqrt[lim]]}, {j, i, TriIndex[lim/tri[[i]]] - 1}]][[2, 1]]]
%t Module[{upto=530000,maxr},maxr=Ceiling[(Sqrt[1+8*Ceiling[upto/3]]-1)/2]; Union[Select[Times@@@Tuples[Rest[Accumulate[Range[maxr]]],2], IntegerQ[ Sqrt[1+8#]]&&#<=upto&]]] (* _Harvey P. Dale_, Jun 12 2012 *)
%Y Cf. A000217 (triangular numbers), A085780 (products of two triangular numbers), A140089 (products of two triangular numbers > 1).
%K nonn
%O 1,1
%A _T. D. Noe_, Apr 06 2011
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