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A286219 Number of partitions of n into parts with an even number of prime divisors (counted with multiplicity).

%I A286219 #10 Feb 16 2025 08:33:44
%S A286219 1,1,1,1,2,2,3,3,4,5,7,7,9,10,13,15,19,20,25,28,34,38,46,50,61,69,81,
%T A286219 89,105,116,137,152,175,194,226,250,288,318,363,403,462,508,577,637,
%U A286219 721,796,900,988,1113,1228,1378,1515,1696,1860,2080,2287,2546,2791,3106,3402,3779
%N A286219 Number of partitions of n into parts with an even number of prime divisors (counted with multiplicity).
%H A286219 Alois P. Heinz, <a href="/A286219/b286219.txt">Table of n, a(n) for n = 0..10000</a>
%H A286219 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>
%H A286219 <a href="/index/Par#part">Index entries for related partition-counting sequences</a>
%F A286219 G.f.: Product_{k>=1} 1/(1 - x^A028260(k)).
%e A286219 a(8) = 4 because we have [6, 1, 1], [4, 4], [4, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1].
%p A286219 with(numtheory):
%p A286219 a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
%p A286219       `if`(bigomega(d)::odd, 0, d), d=divisors(j)), j=1..n)/n)
%p A286219     end:
%p A286219 seq(a(n), n=0..80);  # _Alois P. Heinz_, May 04 2017
%t A286219 nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[EvenQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A286219 Cf. A028260, A087153, A285798, A286218.
%K A286219 nonn
%O A286219 0,5
%A A286219 _Ilya Gutkovskiy_, May 04 2017