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A344446 Number of partitions of n into 3 semiprimes.

%I A344446 #20 Oct 12 2023 15:51:56
%S A344446 0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,1,1,2,1,1,1,3,2,3,2,2,3,3,4,4,2,3,5,
%T A344446 7,5,4,4,5,6,7,7,6,6,7,10,10,7,7,9,10,11,11,13,13,8,12,14,14,13,9,13,
%U A344446 14,16,17,19,15,15,15,22,23,15,14,19,23,23,22,24
%N A344446 Number of partitions of n into 3 semiprimes.
%H A344446 Alois P. Heinz, <a href="/A344446/b344446.txt">Table of n, a(n) for n = 0..20000</a>
%F A344446 a(n) = [x^n y^3] 1/Product_{j>=1} (1-y*x^A001358(j)).
%p A344446 h:= proc(n) option remember; `if`(n=0, 0,
%p A344446      `if`(numtheory[bigomega](n)=2, n, h(n-1)))
%p A344446     end:
%p A344446 b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
%p A344446      `if`(i>n, 0, x*b(n-i, h(min(n-i, i))))+b(n, h(i-1)))), x, 4)
%p A344446     end:
%p A344446 a:= n-> coeff(b(n, h(n)), x, 3):
%p A344446 seq(a(n), n=0..80);
%t A344446 Table[Count[IntegerPartitions[n,{3}],_?(PrimeOmega[#]=={2,2,2}&)],{n,0,80}] (* _Harvey P. Dale_, Oct 12 2023 *)
%Y A344446 Column k=3 of A344447.
%Y A344446 Cf. A001358.
%K A344446 nonn,look
%O A344446 0,19
%A A344446 _Alois P. Heinz_, May 19 2021