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A344255 Number of partitions of n into 8 semiprime parts.

%I A344255 #11 May 15 2025 09:46:25
%S A344255 1,0,1,0,1,1,2,1,2,1,4,3,5,3,5,5,8,8,9,8,12,12,17,16,18,18,22,25,30,
%T A344255 29,33,36,44,45,51,54,59,63,71,78,87,90,99,106,120,124,136,147,157,
%U A344255 166,182,199,216,223,238,259,280,298,314
%N A344255 Number of partitions of n into 8 semiprime parts.
%H A344255 Alois P. Heinz, <a href="/A344255/b344255.txt">Table of n, a(n) for n = 32..10000</a>
%H A344255 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A344255 a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} [Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.
%F A344255 a(n) = [x^n y^8] 1/Product_{j>=1} (1-y*x^A001358(j)). - _Alois P. Heinz_, May 21 2021
%t A344255 h[n_] := h[n] = If[n == 0, 0, If[PrimeOmega[n] == 2, n, h[n-1]]];
%t A344255 b[n_, i_] := b[n, i] = Series[If[n == 0, 1, If[i < 1, 0, If[i > n, 0, x*b[n-i, h[Min[n-i, i]]]]+b[n, h[i-1]]]], {x, 0, 9}];
%t A344255 a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 8}];
%t A344255 Table[a[n], {n, 32, 120}] (* _Jean-François Alcover_, May 15 2025, after _Alois P. Heinz_ in A344257 *)
%Y A344255 Cf. A001222 (Omega), A001358.
%Y A344255 Column k=8 of A344447.
%K A344255 nonn
%O A344255 32,7
%A A344255 _Wesley Ivan Hurt_, May 12 2021