A188660 - OEIS

%I #21 Sep 28 2024 07:10:16

%S 12,72,240,420,600,1260,2352,4032,6480,7140,9900,14280,14520,20592,

%T 28392,38220,46872,50400,65280,78120,83232,104652,123552,129960,

%U 159600,194040,233772,279312,291060,331200,390000,456300,485112,530712,609180,613872,699732,706440,809100,852852,922560

%N Oblong numbers that are the product of two oblong numbers.

%C Breiteig writes about the finding these numbers, but does not list these numbers as a sequence. The problem can be extended to any polygonal number: for example, when is a pentagonal number the product of two pentagonal numbers? See A188630 and A188663 for the triangular and pentagonal cases.

%C As shown in the example, the product of consecutive oblong numbers is also oblong: oblong(n) * oblong(n+1) = oblong(n*(n+2)).

%H Donovan Johnson, <a href="/A188660/b188660.txt">Table of n, a(n) for n = 1..5000</a>

%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.

%e 240 = 12 * 20; that is, oblong(15) = oblong(3) * oblong(4).

%t OblongQ[n_] := IntegerQ[Sqrt[1 + 4 n]]; OblongIndex[n_] := Floor[(-1 + Sqrt[1 + 4*n])/2]; lim = 10^6; nMax = OblongIndex[lim/2]; obl = Table[n (n + 1), {n, nMax}]; Union[Reap[Do[num = obl[[i]]*obl[[j]]; If[OblongQ[num], Sow[num]], {i, OblongIndex[Sqrt[lim]]}, {j, i, OblongIndex[lim/obl[[i]]]}]][[2, 1]]]

%Y Cf. A002378 (oblong numbers), A188630, A188663, A374374 (more than 2 factors allowed).

%K nonn

%O 1,1

%A _T. D. Noe_, Apr 07 2011