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A188630 Triangular numbers that are the product of two triangular numbers greater than 1.

%I A188630 #39 Jan 17 2025 19:38:03
%S A188630 36,45,210,630,780,990,1540,2850,3570,4095,4851,8778,11781,15400,
%T A188630 17955,19110,21528,25200,26565,26796,33930,37128,40755,43956,61425,
%U A188630 61776,70125,79800,105570,113050,122265,145530,176715,189420,192510,246753,270480,303810,349866,437580,500500,526851
%N A188630 Triangular numbers that are the product of two triangular numbers greater than 1.
%C A188630 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 A188630 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 A188630 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 A188630 Donovan Johnson, <a href="/A188630/b188630.txt">Table of n, a(n) for n = 1..4082</a>
%H A188630 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.
%e A188630 210 = T(20) = 10 * 21 = T(4) * T(6).
%p A188630 A188630 := proc(limit) local t,E,n,k,c,b,ist; E:=NULL;
%p A188630 t := proc(n) option remember; iquo(n*(n+1), 2) end;
%p A188630 ist := proc(n) option remember; n = t(floor(sqrt(2*n))) end;
%p A188630 for n from 2 do
%p A188630     c := t(n); if c > limit then break fi;
%p A188630     for k from 2 do
%p A188630         b := c*t(k); if b > limit then break fi;
%p A188630         if ist(b) then E := E, b fi;
%p A188630 od od; sort({E}) end:
%p A188630 A188630(200000); # _Peter Luschny_, Dec 21 2017
%t A188630 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 A188630 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 A188630 Cf. A000217 (triangular numbers), A085780 (products of two triangular numbers), A140089 (products of two triangular numbers > 1).
%Y A188630 Subsequence of A068143 (more than 2 factors allowed).
%Y A188630 Cf. A001571, A061278, A188660, A188663.
%Y A188630 See also A379609.
%K A188630 nonn
%O A188630 1,1
%A A188630 _T. D. Noe_, Apr 06 2011