A286219 - OEIS

%I #9 May 07 2017 16:48:59

%S 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 89,105,116,137,152,175,194,226,250,288,318,363,403,462,508,577,637,

%U 721,796,900,988,1113,1228,1378,1515,1696,1860,2080,2287,2546,2791,3106,3402,3779

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

%H Alois P. Heinz, <a href="/A286219/b286219.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>

%H <a href="/index/Par#part">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=1} 1/(1 - x^A028260(k)).

%e 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 with(numtheory):

%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(

%p       `if`(bigomega(d)::odd, 0, d), d=divisors(j)), j=1..n)/n)

%p     end:

%p seq(a(n), n=0..80);  # _Alois P. Heinz_, May 04 2017

%t nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[EvenQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A028260, A087153, A285798, A286218.

%K nonn

%O 0,5

%A _Ilya Gutkovskiy_, May 04 2017