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Pentagonal numbers that are products of smaller pentagonal numbers.
3

%I #5 Jul 07 2024 13:51:18

%S 1,10045,11310,20475,52360,197472,230300,341055,367290,836640,2437800,

%T 2939300,3262700,4048352,4268110,4293450,4619160,4816000,5969040,

%U 6192520,6913340,6997320,8531145,10933650,12397000,16008300,18573282,18816875,21430710,24383520

%N Pentagonal numbers that are products of smaller pentagonal numbers.

%C 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.

%e 1 is a term because it is a pentagonal number and equals the empty product.

%e 10045 is a term because it is a pentagonal number and equals the product of the pentagonal numbers 35 and 287.

%e 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.)

%Y Row n=5 of A374370.

%Y A188663 is a subsequence (only 2 factors allowed).

%Y Cf. A000326.

%K nonn

%O 1,2

%A _Pontus von Brömssen_, Jul 07 2024