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

1,1

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)).

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.

Example

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

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]]]

Crossrefs

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

Sequence in context: A379833 A199531 A374374 * A047928 A300847 A235870

Adjacent sequences: A188657 A188658 A188659 * A188661 A188662 A188663

Keyword

nonn

Author

T. D. Noe, Apr 07 2011

Status

approved