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A286218 Number of partitions of n into parts with an odd number of prime divisors (counted with multiplicity).

%I A286218 #14 Feb 16 2025 08:33:44
%S A286218 1,0,1,1,1,2,2,3,4,4,6,7,9,11,13,16,19,23,28,33,40,46,55,65,76,89,104,
%T A286218 121,141,163,190,219,253,290,334,383,439,502,573,653,744,845,961,1089,
%U A286218 1234,1395,1576,1780,2007,2259,2544,2856,3209,3598,4033,4516,5051,5644,6304,7033,7843
%N A286218 Number of partitions of n into parts with an odd number of prime divisors (counted with multiplicity).
%H A286218 Alois P. Heinz, <a href="/A286218/b286218.txt">Table of n, a(n) for n = 0..10000</a>
%H A286218 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>
%H A286218 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A286218 G.f.: Product_{k>=1} 1/(1 - x^A026424(k)).
%e A286218 a(8) = 4 because we have [8], [5, 3], [3, 3, 2] and [2, 2, 2, 2].
%p A286218 with(numtheory):
%p A286218 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A286218       `if`(bigomega(d)::odd, d, 0), d=divisors(j)), j=1..n)/n)
%p A286218     end:
%p A286218 seq(a(n), n=0..80);  # _Alois P. Heinz_, May 04 2017
%t A286218 nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[OddQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A286218 Cf. A001156, A026424, A285799, A286219.
%K A286218 nonn
%O A286218 0,6
%A A286218 _Ilya Gutkovskiy_, May 04 2017