A327641 - OEIS

%I #11 Jan 28 2025 12:26:11

%S 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,

%T 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,

%U 38,14,42,47,2,742,2,50,18,2,44,5257,2,29,50,5066,2,8,2,59,14

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

%H Antti Karttunen, <a href="/A327641/b327641.txt">Table of n, a(n) for n = 0..20000</a>

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

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

%o (Magma) [1] cat [#RestrictedPartitions(n,{d:d in Divisors(n)|IsSquarefree(n div d)}):n in [1..75]]; // _Marius A. Burtea_, Sep 20 2019

%o (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

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

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Sep 20 2019