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A280285 Number of partitions of n into odd composite numbers (A071904).

%I A280285 #12 Feb 16 2025 08:33:38
%S A280285 1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,1,1,0,2,0,0,2,0,0,2,
%T A280285 1,1,3,0,0,3,1,0,4,1,1,5,1,0,5,2,2,6,2,1,8,3,1,8,3,2,11,3,2,12,5,4,13,
%U A280285 5,3,16,8,4,18,7,6,22,9,7,24,12,9,28,12,9,33,18,11,36,18,14,45,22,16,48,26,22,54,29,23,66,38
%N A280285 Number of partitions of n into odd composite numbers (A071904).
%H A280285 Alois P. Heinz, <a href="/A280285/b280285.txt">Table of n, a(n) for n = 0..10000</a>
%H A280285 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompositeNumber.html">Composite Number</a>
%H A280285 <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F A280285 G.f.: ((1 - x)/(1 - x^2))*Product_{k>=1} (1 - x^(2*k))*(1 - x^prime(k))/(1 - x^k).
%e A280285 a(36) = 3 because we have [27, 9], [21, 15] and [9, 9, 9, 9].
%p A280285 a:= proc(n) option remember; `if`(n=0, 1, add(add(
%p A280285       `if`(d>1 and d::odd and not isprime(d), d, 0),
%p A280285        d=numtheory[divisors](j))*a(n-j), j=1..n)/n)
%p A280285     end:
%p A280285 seq(a(n), n=0..100);  # _Alois P. Heinz_, Dec 31 2016
%t A280285 nmax = 100; CoefficientList[Series[(1 - x)/(1 - x^2) Product[(1 - x^(2 k)) (1 - x^Prime[k])/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A280285 Cf. A002095, A002808, A023895, A071904, A204389, A280287.
%K A280285 nonn
%O A280285 0,28
%A A280285 _Ilya Gutkovskiy_, Dec 31 2016