A352063 Number of ordered factorizations of 2*n + 1 into odd factors > 1.

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 4, 1, 1, 3, 3, 1, 3, 1, 1, 8, 1, 2, 3, 1, 3, 3, 1, 1, 8, 3, 1, 3, 1, 1, 8, 3, 1, 8, 1, 3, 3, 1, 3, 3, 3, 1, 8, 1, 1, 13, 1, 1, 3, 1, 3, 8, 3, 2, 3, 4, 1, 3, 1, 3, 20, 1, 1, 3, 3, 3, 8, 1, 1, 8, 3, 1, 3, 3, 1, 13, 1, 2, 8, 1, 8, 3, 1, 1, 3, 3, 3, 20, 1, 1, 13, 1, 1, 3, 3, 3, 8, 3, 1

Offset: 0

Author

Ilya Gutkovskiy, Mar 04 2022

Keywords

• nonn

Comments

Also number of perfect partitions of 2*n.

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20000
• Eric Weisstein's World of Mathematics, Perfect Partition
• Eric Weisstein's World of Mathematics, Ordered Factorization

Crossrefs

Cf. A002033, A058696, A064216, A074206, A108503, A340101, A349907, A378222.

Programs

Mathematica

A074206[1] = 1; A074206[n_] := A074206[n] = Sum[If[d < n, A074206[d], 0], {d, Divisors[n]}]; a[n_] := A074206[2 n + 1]; Table[a[n], {n, 0, 90}]

PARI

A074206(n) = if(n>1, sumdiv(n, i, if(iA074206(i))), n); \\ From A074206
A352063(n) = A074206(1+n+n); \\ Antti Karttunen, Nov 24 2024

Formula

a(n) = A002033(2*n) = A074206(2*n+1).

a(n) = A378222(2*n+1) = A074206(A064216(1+n)). - Antti Karttunen, Nov 24 2024

Extensions

More terms from Antti Karttunen, Nov 24 2024