A374374 -id:A374374 - cp's OEIS frontend

A068143 Triangular numbers which are products of triangular numbers larger than 1.

36, 45, 210, 300, 378, 630, 780, 990, 1485, 1540, 2850, 3240, 3570, 4095, 4851, 4950, 5460, 8778, 9180, 11781, 15400, 17955, 19110, 21528, 25200, 26565, 26796, 31878, 33930, 37128, 37950, 39060, 40755, 43956, 52650, 61425, 61776, 70125, 79800, 82215, 97020

Offset: 1

Amarnath Murthy, Feb 23 2002

nonn

			210 is a term as 210 = 21*10 both triangular numbers.
300 is also a term since 300 = 3*10*10.

Pontus von Brömssen, Table of n, a(n) for n = 1..3597 (all terms below 10^12).

Cf. A000217 (triangular numbers), A374374.
Except the first term, row n=3 of A374370 and row n=2 of A374498.
A188630 is a subsequence (only 2 factors allowed).

Corrected and extended by Jon E. Schoenfield, Jul 23 2006

a(38)-a(41) from Pontus von Brömssen, Jul 02 2024

A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

Original entry on oeis.org

0, 3, 7, 13, 17, 18, 21, 31, 38, 43, 47, 57, 68, 73, 91, 99, 111, 117, 123, 132, 133, 157, 183, 211, 241, 242, 253, 255, 268, 273, 293, 302, 307, 313, 322, 327, 343, 381, 413, 421, 438, 443, 463, 487, 507, 515, 553, 557, 577, 593, 601, 651, 693, 697, 703, 707

Offset: 1

Pontus von Brömssen, Jun 19 2023

nonn

For the corresponding sequence for numbers of the form k^3+1 instead of k^2+1, the only terms known to me are 0 and 26, with 26^3+1 = (2^3+1)^2*(6^3+1).

			0 is a term because 0^2+1 = 1 equals the empty product.
3 is a term because 3^2+1 = 10 = 2*5 = (1^2+1)*(2^2+1).
38 is a term because 38^2+1 = 1445 = 5*17*17 = (2^2+1)*(4^2+1)^2. (This is the first term that requires more than two factors.)

David Trimas, Table of n, a(n) for n = 1..2260

Sequences that list those terms (or their indices or some other key) of a given sequence that are products of smaller terms of the same sequence (in other words, the nonprimitive terms of the multiplicative closure of the sequence):
  A018252 (A000027),
  A034878 (A000142),
  A068143 (A000217),
  A363492 (A000041),
  A363634 (A000959),
  A363635 (A003309),
  this sequence (A002522),
  A363637 (A005563),
  A363638 (A008864),
  A363750 (A006093),
  A364151 (A000292),
  A374372 (A000326),
  A374373 (A000384),
  A374374 (A002378),
  A374375 (A007531),
  A374500 (A000330),
  A374501 (A002411),
  A374502 (A002412).

g[lst_, p_] :=
  Module[{t, i, j},
   Union[Flatten[Table[t = lst[[i]]; t[[j]] = p*t[[j]];
      Sort[t], {i, Length[lst]}, {j, Length[lst[[i]]]}], 1],
    Table[Sort[Append[lst[[i]], p]], {i, Length[lst]}]]];
multPartition[n_] :=
  Module[{i, j, p, e, lst = {{}}}, {p, e} =
    Transpose[FactorInteger[n]];
   Do[lst = g[lst, p[[i]]], {i, Length[p]}, {j, e[[i]]}]; lst];
output = Join[{0}, Flatten[Position[Table[
     test = Sqrt[multPartition[n^2 + 1][[2 ;; All]] - 1];
     Count[AllTrue[#, IntegerQ] & /@ test, True] > 0
     , {n, 707}], True]]]
(* David Trimas, Jul 23 2023 *)

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]]]

A374375 Positive numbers of the form k(k+1)(k+2) that are products of smaller numbers of that same form.

Original entry on oeis.org

720, 262080, 43243200, 85765680, 14366626560, 27680637600, 8916100427520, 2871098559070560, 5720836667515200, 20123426048544000, 924491486191094640, 297683700627082714560

Offset: 1

Pontus von Brömssen, Jul 07 2024

All terms are divisible by 36, because the number k*(k+1)*(k+2) is always divisible by 6 so a product of at least 2 such factors is divisible by 36. The first 12 terms are even divisible by 720.
a(13) > 4.5*10^22 if it exists. - David A. Corneth, Jul 12 2024
b(F(2*k)^2-1) is a term for all k >= 2, where b(k) = k*(k+1)*(k+2) = A007531(k+2) and F = A000045 is the Fibonacci sequence, because b(F(2*k)^2-1) = b(F(2*k-1)-1)*b(F(2*k+1)-1). In particular, a(13) <= b(F(20)^2-1) = 95853241822852400000400. - Pontus von Brömssen, Jul 13 2024

			With b(k) = k*(k+1)*(k+2) = A007531(k+2), we have the following factorizations of the first 12 terms:
                    720 =       b(8) = 6*120 = b(1)*b(4);
                 262080 =      b(63) = 120*2184 = b(4)*b(12);
               43243200 =     b(350) = 120*210*1716 = b(4)*b(5)*b(11);
               85765680 =     b(440) = 2184*39270 = b(12)*b(33);
            14366626560 =    b(2430) = 24*60*1716*5814 = b(2)*b(3)*b(11)*b(17);
            27680637600 =    b(3024) = 39270*704880 = b(33)*b(88);
          8916100427520 =   b(20735) = 704880*12649104 = b(88)*b(232);
       2871098559070560 =  b(142128) = 12649104*226980390 = b(232)*b(609);
       5720836667515200 =  b(178848) = 6*210*373176*12166770 = b(1)*b(5)*b(71)*b(229);
      20123426048544000 =  b(271998) = 6*210*328440*48626760 = b(1)*b(5)*b(68)*b(364);
     924491486191094640 =  b(974168) = 226980390*4073001576 = b(609)*b(1596);
  297683700627082714560 = b(6677055) = 4073001576*73087057560 = b(1596)*b(4180).

David A. Corneth, PARI program

Cf. A000045, A007531, A364151, A374374.

a(8)-a(11) from Michael S. Branicky, Jul 07 2024

a(12) from David A. Corneth, Jul 12 2024

cp's OEIS Frontend

A068143 Triangular numbers which are products of triangular numbers larger than 1.

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A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

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

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A374375 Positive numbers of the form k(k+1)(k+2) that are products of smaller numbers of that same form.

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cp's OEIS Frontend

A068143 Triangular numbers which are products of triangular numbers larger than 1.

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Author

Keywords

Examples

Links

Crossrefs

Extensions

A363636 Indices of numbers of the form k^2+1, k >= 0, that can be written as a product of smaller numbers of that same form.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A374375 Positive numbers of the form k*(k+1)*(k+2) that are products of smaller numbers of that same form.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A374375 Positive numbers of the form k(k+1)(k+2) that are products of smaller numbers of that same form.