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

0,5

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

Formula

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

Example

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].

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]

Crossrefs

Cf. A028260, A087153, A285798, A286218.

Sequence in context: A130083 A369787 A363994 * A117357 A029020 A035380

Adjacent sequences: A286216 A286217 A286218 * A286220 A286221 A286222

Keyword

nonn

Author

Ilya Gutkovskiy, May 04 2017

Status

approved