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Number of partitions of n into distinct parts with an even number of prime divisors (counted with multiplicity).
3

%I #9 Feb 16 2025 08:33:44

%S 1,1,0,0,1,1,1,1,0,1,3,2,0,1,3,4,4,2,1,4,6,5,4,4,6,10,10,6,6,10,13,14,

%T 11,9,14,21,21,17,17,23,31,31,25,25,33,41,43,39,38,50,61,60,56,58,68,

%U 83,87,79,82,99,115,121,118,118,139,163,164,157,165,189,216,228,221,229,265,296

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

%H Amiram Eldar, <a href="/A286223/b286223.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeFactor.html">Prime Factor</a>

%H <a href="/index/Par#part">Index entries for related partition-counting sequences</a>

%F G.f.: Product_{k>=1} (1 + x^A028260(k)).

%e a(10) = 3 because we have [10], [9, 1] and [6, 4].

%t nmax = 75; CoefficientList[Series[Product[1 + Boole[EvenQ[PrimeOmega[k]]] x^k, {k, 1, nmax}], {x, 0, nmax}], x]

%Y Cf. A028260, A286219, A286221, A286222.

%K nonn,changed

%O 0,11

%A _Ilya Gutkovskiy_, May 04 2017