A374372 Pentagonal numbers that are products of smaller pentagonal numbers.

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

Author

Pontus von Brömssen, Jul 07 2024

Keywords

• nonn

Comments

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.

Examples

			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.)
		

Crossrefs

Row n=5 of A374370.

A188663 is a subsequence (only 2 factors allowed).

Cf. A000326.