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A286219
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Number of partitions of n into parts with an even number of prime divisors (counted with multiplicity).
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5
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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
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OFFSET
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0,5
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LINKS
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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
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FORMULA
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G.f.: Product_{k>=1} 1/(1 - x^A028260(k)).
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EXAMPLE
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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].
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MAPLE
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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
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MATHEMATICA
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nmax = 60; CoefficientList[Series[Product[1/(1 - Boole[EvenQ[PrimeOmega[k]]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
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CROSSREFS
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Cf. A028260, A087153, A285798, A286218.
Sequence in context: A064650 A174619 A130083 * A117357 A029020 A035380
Adjacent sequences: A286216 A286217 A286218 * A286220 A286221 A286222
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KEYWORD
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nonn
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AUTHOR
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Ilya Gutkovskiy, May 04 2017
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STATUS
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approved
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