A374500 -id:A374500 - cp's OEIS frontend

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.

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 *)

A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

Original entry on oeis.org

1, 36, 1, 45, 560, 1, 210, 19600, 4900, 1, 300, 43680, 513590, 56448, 1, 378, 45760, 333833500, 127008, 4750, 1, 630, 893200, 711410700, 259200, 1926049000, 58372180608, 1

Offset: 2

Pontus von Brömssen, Jul 09 2024

If there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

The first term in each row is 1, because 1 is an n-gonal pyramidal number for every n and it equals the empty product.

			Array begins:
  n\k| 1     2          3            4
  ---+--------------------------------
  2  | 1    36         45          210
  3  | 1   560      19600        43680
  4  | 1  4900     513590    333833500
  5  | 1 56448     127008       259200
  6  | 1  4750 1926049000 655578709500

Cf. A080851, A374370, A374499 (second column).

Rows: A068143 (n=2 except the first term), A364151 (n=3), A374500 (n=4), A374501 (n=5), A374502 (n=6).

A374501 Pentagonal pyramidal numbers that are products of smaller pentagonal pyramidal numbers.

Original entry on oeis.org

1, 56448, 127008, 259200, 5644800, 31840200, 42688800, 60766200, 116493300, 130662720, 193179168, 442828800, 499000500, 897544800, 917632800, 3624409800, 6914880000, 13831171200, 15410656800, 31246000128, 53936416800, 64024732800, 72945774720, 88957620000

Offset: 1

Pontus von Brömssen, Jul 09 2024

nonn

			1 is a term because it is a pentagonal pyramidal number and equals the empty product.
56448 is a term because it is a pentagonal pyramidal number and equals the product of the pentagonal pyramidal numbers 196 and 288.
127008 is a term because it is a pentagonal pyramidal number and equals the product of the pentagonal pyramidal numbers 6, 6, 18, and 196.

Row n=5 of A374498.

Cf. A002411, A374500, A374502.

A374502 Hexagonal pyramidal numbers that are products of smaller hexagonal pyramidal numbers.

Original entry on oeis.org

1, 4750, 1926049000, 655578709500, 9126464328696330

Offset: 1

Pontus von Brömssen, Jul 09 2024

a(6) > 10^19 (if it exists). - Pontus von Brömssen, Jul 14 2024

			1 is a term because it is a hexagonal pyramidal number and equals the empty product.
4750 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 50 and 95.
1926049000 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 9500 and 202742.
655578709500 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 50, 31746, and 413015.
9126464328696330 is a term because it is a hexagonal pyramidal number and equals the product of the hexagonal pyramidal numbers 413015 and 22097174022.

Row n=6 of A374498.

Cf. A002412, A374500, A374501.

a(5) from Michael S. Branicky, Jul 09 2024

cp's OEIS Frontend

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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A374498 Square array read by antidiagonals: row n lists n-gonal pyramidal numbers that are products of smaller n-gonal pyramidal numbers.

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A374501 Pentagonal pyramidal numbers that are products of smaller pentagonal pyramidal numbers.

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A374502 Hexagonal pyramidal numbers that are products of smaller hexagonal pyramidal numbers.

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