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

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, 89, 105, 116, 137, 152, 175, 194, 226, 250, 288, 318, 363, 403, 462, 508, 577, 637, 721, 796, 900, 988, 1113, 1228, 1378, 1515, 1696, 1860, 2080, 2287, 2546, 2791, 3106, 3402, 3779

Offset: 0

Author

Ilya Gutkovskiy, May 04 2017

Keywords

• nonn

Examples

			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].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Prime Factor
• Index entries for related partition-counting sequences

Crossrefs

Cf. A028260, A087153, A285798, A286218.

Programs

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(bigomega(d)::odd, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..80);  # Alois P. Heinz, May 04 2017

Mathematica

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

Formula

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