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A344256 Number of partitions of n into 9 semiprime parts.

%I A344256 #14 May 15 2025 09:46:29
%S A344256 1,0,1,0,1,1,2,1,2,1,4,3,5,3,5,5,8,8,10,8,12,13,18,16,19,19,24,27,31,
%T A344256 31,37,38,46,50,57,58,65,71,80,86,96,102,115,119,134,146,160,167,181,
%U A344256 197,217,232,252,269,290,306,333,364,387,407,434,474,512,541
%N A344256 Number of partitions of n into 9 semiprime parts.
%H A344256 Alois P. Heinz, <a href="/A344256/b344256.txt">Table of n, a(n) for n = 36..10000</a>
%H A344256 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A344256 a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} [Omega(q) = Omega(p) = Omega(o) = Omega(m) = Omega(l) = Omega(k) = Omega(j) = Omega(i) = Omega(n-i-j-k-l-m-o-p-q) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.
%F A344256 a(n) = [x^n y^9] 1/Product_{j>=1} (1-y*x^A001358(j)). - _Alois P. Heinz_, May 21 2021
%t A344256 h[n_] := h[n] = If[n == 0, 0,If[PrimeOmega[n] == 2, n, h[n-1]]];
%t A344256 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, 10}];
%t A344256 a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 9}];
%t A344256 Table[a[n], {n, 36, 120}] (* _Jean-François Alcover_, May 15 2025, after _Alois P. Heinz_ in A344257 *)
%Y A344256 Cf. A001222 (Omega), A001358.
%Y A344256 Column k=9 of A344447.
%K A344256 nonn
%O A344256 36,7
%A A344256 _Wesley Ivan Hurt_, May 13 2021
%E A344256 More terms from _Alois P. Heinz_, May 18 2021