A344255 Number of partitions of n into 8 semiprime parts.

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, 29, 33, 36, 44, 45, 51, 54, 59, 63, 71, 78, 87, 90, 99, 106, 120, 124, 136, 147, 157, 166, 182, 199, 216, 223, 238, 259, 280, 298, 314

Offset: 32

Author

Wesley Ivan Hurt, May 12 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 32..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A001222 (Omega), A001358.

Column k=8 of A344447.

Programs

Mathematica

h[n_] := h[n] = If[n == 0, 0, If[PrimeOmega[n] == 2, n, h[n-1]]];
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}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 8}];
Table[a[n], {n, 32, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz in A344257 *)

Formula

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.

a(n) = [x^n y^8] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021