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

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, 121, 141, 163, 190, 219, 253, 290, 334, 383, 439, 502, 573, 653, 744, 845, 961, 1089, 1234, 1395, 1576, 1780, 2007, 2259, 2544, 2856, 3209, 3598, 4033, 4516, 5051, 5644, 6304, 7033, 7843

Offset

0,6

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^A026424(k)).

Example

a(8) = 4 because we have [8], [5, 3], [3, 3, 2] and [2, 2, 2, 2].

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
      `if`(bigomega(d)::odd, d, 0), 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[OddQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A001156, A026424, A285799, A286219.

Sequence in context: A053281 A339395 A228117 * A094997 A173673 A330715

Adjacent sequences: A286215 A286216 A286217 * A286219 A286220 A286221

Keyword

nonn

Author

Ilya Gutkovskiy, May 04 2017

Status

approved