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A344257 Number of partitions of n into 10 semiprime parts.

%I A344257 #22 May 15 2025 08:05:53
%S A344257 1,0,1,0,1,1,2,1,2,1,4,3,5,3,5,5,8,8,10,8,13,13,18,17,20,19,25,28,33,
%T A344257 33,38,40,50,52,59,63,71,75,86,94,105,110,124,131,150,159,174,189,205,
%U A344257 217,242,264,288,303,327,354,388,414,443,476,511,547,594,641
%N A344257 Number of partitions of n into 10 semiprime parts.
%H A344257 Alois P. Heinz, <a href="/A344257/b344257.txt">Table of n, a(n) for n = 40..10000</a>
%H A344257 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A344257 a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} [Omega(r) = 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-r) = 2], where Omega is the number of prime factors with multiplicity (A001222) and [ ] is the (generalized) Iverson bracket.
%F A344257 a(n) = [x^n y^10] 1/Product_{j>=1} (1-y*x^A001358(j)). - _Alois P. Heinz_, May 19 2021
%p A344257 h:= proc(n) option remember; `if`(n=0, 0,
%p A344257      `if`(numtheory[bigomega](n)=2, n, h(n-1)))
%p A344257     end:
%p A344257 b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,
%p A344257      `if`(i>n, 0, x*b(n-i, h(min(n-i, i))))+b(n, h(i-1)))), x, 11)
%p A344257     end:
%p A344257 a:= n-> coeff(b(n, h(n)), x, 10):
%p A344257 seq(a(n), n=40..120);  # _Alois P. Heinz_, May 26 2021
%t A344257 h[n_] := h[n] = If[n == 0, 0, If[PrimeOmega[n] == 2, n, h[n-1]]];
%t A344257 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, 11}];
%t A344257 a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 10}];
%t A344257 Table[a[n], {n, 40, 120}] (* _Jean-François Alcover_, May 15 2025, after _Alois P. Heinz_ *)
%Y A344257 Cf. A001222 (Omega), A001358.
%Y A344257 Column k=10 of A344447.
%K A344257 nonn
%O A344257 40,7
%A A344257 _Wesley Ivan Hurt_, May 13 2021
%E A344257 a(83)-a(103) from _Alois P. Heinz_, May 18 2021