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A286219
Number of partitions of n into parts with an even number of prime divisors (counted with multiplicity).
5
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
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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 04 2017
STATUS
approved