A327641 Number of partitions of n into divisors d of n such that n/d is squarefree.

1, 1, 2, 2, 2, 2, 8, 2, 2, 2, 11, 2, 8, 2, 14, 14, 2, 2, 8, 2, 11, 18, 20, 2, 8, 2, 23, 2, 14, 2, 742, 2, 2, 26, 29, 26, 8, 2, 32, 30, 11, 2, 1654, 2, 20, 14, 38, 2, 8, 2, 11, 38, 23, 2, 8, 38, 14, 42, 47, 2, 742, 2, 50, 18, 2, 44, 5257, 2, 29, 50, 5066, 2, 8, 2, 59, 14

Offset: 0

Author

Ilya Gutkovskiy, Sep 20 2019

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A008683, A018818, A225244, A225245, A246655 (positions of 2's).

Programs

Magma

[1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)|IsSquarefree(n div d)}):n in [1..75]]; // Marius A. Burtea, Sep 20 2019

Mathematica

a[n_] := SeriesCoefficient[Product[1/(1 - MoebiusMu[n/d]^2 x^d), {d, Divisors[n]}], {x, 0, n}]; Table[a[n], {n, 0, 75}]

PARI

A327641(n) = if(!n, 1, my(p = Ser(1, 'x, 1+n)); fordiv(n, d, if(issquarefree(n/d), p /= (1 - 'x^d))); polcoef(p, n)); \\ Antti Karttunen, Jan 28 2025

Formula

a(n) = [x^n] Product_{d|n} 1 / (1 - mu(n/d)^2 * x^d).