A188660 Oblong numbers that are the product of two oblong numbers.

12, 72, 240, 420, 600, 1260, 2352, 4032, 6480, 7140, 9900, 14280, 14520, 20592, 28392, 38220, 46872, 50400, 65280, 78120, 83232, 104652, 123552, 129960, 159600, 194040, 233772, 279312, 291060, 331200, 390000, 456300, 485112, 530712, 609180, 613872, 699732, 706440, 809100, 852852, 922560

Offset: 1

Author

T. D. Noe, Apr 07 2011

Keywords

• nonn

Comments

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.

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

Examples

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

Links

• Donovan Johnson, Table of n, a(n) for n = 1..5000
• Trygve Breiteig, When is the product of two oblong numbers another oblong?, Math. Mag. 73 (2000), 120-129.

Crossrefs

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

Programs

Mathematica

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]]]