

A188630


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


18



36, 45, 210, 630, 780, 990, 1540, 2850, 3570, 4095, 4851, 8778, 11781, 15400, 17955, 19110, 21528, 25200, 26565, 26796, 33930, 37128, 40755, 43956, 61425, 61776, 70125, 79800, 105570, 113050, 122265, 145530, 176715, 189420, 192510, 246753, 270480, 303810, 349866, 437580, 500500, 526851
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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?
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.
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.


LINKS

Donovan Johnson, Table of n, a(n) for n = 1..4082
Trygve Breiteig, When is the product of two oblong numbers another oblong?, Math. Mag. 73 (2000), 120129.


EXAMPLE

210 = T(20) = 10 * 21 = T(4) * T(6).


MAPLE

A188630 := proc(limit) local t, E, n, k, c, b, ist; E:=NULL;
t := proc(n) option remember; iquo(n*(n+1), 2) end;
ist := proc(n) option remember; n = t(floor(sqrt(2*n))) end;
for n from 2 do
c := t(n); if c > limit then break fi;
for k from 2 do
b := c*t(k); if b > limit then break fi;
if ist(b) then E := E, b fi;
od od; sort({E}) end:
A188630(200000); # Peter Luschny, Dec 21 2017


MATHEMATICA

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]]]
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 *)


CROSSREFS

Cf. A000217 (triangular numbers), A085780 (products of two triangular numbers), A140089 (products of two triangular numbers > 1).
Sequence in context: A144291 A068143 A264961 * A167310 A083674 A160063
Adjacent sequences: A188627 A188628 A188629 * A188631 A188632 A188633


KEYWORD

nonn


AUTHOR

T. D. Noe, Apr 06 2011


STATUS

approved



