A188663 -id:A188663 - cp's OEIS frontend

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

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

Offset: 1

T. D. Noe, Apr 06 2011

nonn

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.

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

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), 120-129.

Cf. A000217 (triangular numbers), A085780 (products of two triangular numbers), A140089 (products of two triangular numbers > 1).
Subsequence of A068143 (more than 2 factors allowed).
Cf. A001571, A061278, A188660, A188663.
See also A379609.

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

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 *)

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

Original entry on oeis.org

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

T. D. Noe, Apr 07 2011

nonn

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)).

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

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.

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

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]]]

A374372 Pentagonal numbers that are products of smaller pentagonal numbers.

Original entry on oeis.org

1, 10045, 11310, 20475, 52360, 197472, 230300, 341055, 367290, 836640, 2437800, 2939300, 3262700, 4048352, 4268110, 4293450, 4619160, 4816000, 5969040, 6192520, 6913340, 6997320, 8531145, 10933650, 12397000, 16008300, 18573282, 18816875, 21430710, 24383520

Offset: 1

Pontus von Brömssen, Jul 07 2024

nonn

There are infinitely many terms where the corresponding product has two factors. This can be seen by solving the equation A000326(x)=A000326(y)*A000326(z) for a fixed z for which a solution exists, leading to a generalized Pell equation. For example, z = 5 leads to the solutions (x,y) = (82,14), (1649982,278898), (33266933642,5623138102), ..., corresponding to the terms A000326(82) = 10045, A000326(1649982) = 4083660075495, A000326(33266933642) = 1660033310895213609425, ... in the sequence.

			1 is a term because it is a pentagonal number and equals the empty product.
10045 is a term because it is a pentagonal number and equals the product of the pentagonal numbers 35 and 287.
20475 is a term because it is a pentagonal number and equals the product of the pentagonal numbers 5, 35, and 117. (This is the first term that requires more than two factors.)

Row n=5 of A374370.
A188663 is a subsequence (only 2 factors allowed).
Cf. A000326.

A212615 Least k > 1 such that the product pen(n) * pen(k) is pentagonal, or zero if no such k exists, where pen(k) is the k-th pentagonal number.

Original entry on oeis.org

2, 39, 2231, 40, 14, 94974, 47, 212, 1071, 477, 124, 261, 15120, 5, 180, 375638, 2413, 22, 4270831, 924, 278, 18, 126, 33510, 355, 376, 9047610, 37313170, 1533015, 7315, 1687018, 520, 363155, 8827, 13514, 11701449166, 670, 3290, 2, 4, 817, 31175067

Offset: 1

T. D. Noe, Jun 07 2012

nonn

That is, pen(k) = k*(3k-1)/2.

			For n = 2, pen(n) = 5 and the first k is 39 because pen(39) = 2262 and 5*2262 = 11310 which is the 87th pentagonal number.

Cf. A188663 (pentagonal numbers that are pen(x) * pen(y) for some x,y > 1).
Cf. A212614 (similar sequence for triangular numbers).
Cf. A000326 (pentagonal numbers).

kMax = 10^7; PentagonalQ[n_] := IntegerQ[(1 + Sqrt[1 + 24*n])/6]; Table[t = n*(3*n - 1)/2; k = 2; While[t2 = k*(3*k - 1)/2; k < kMax && ! PentagonalQ[t*t2], k++]; If[k == kMax, 0, k], {n, 15}]

a(25) corrected and a(28)-a(42) from Donovan Johnson, Feb 08 2013

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

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

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A374372 Pentagonal numbers that are products of smaller pentagonal numbers.

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A212615 Least k > 1 such that the product pen(n) * pen(k) is pentagonal, or zero if no such k exists, where pen(k) is the k-th pentagonal number.

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