A344256 Number of partitions of n into 9 semiprime parts.

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, 31, 37, 38, 46, 50, 57, 58, 65, 71, 80, 86, 96, 102, 115, 119, 134, 146, 160, 167, 181, 197, 217, 232, 252, 269, 290, 306, 333, 364, 387, 407, 434, 474, 512, 541

Offset

36,7

Links

Alois P. Heinz, Table of n, a(n) for n = 36..10000

Index entries for sequences related to partitions

Formula

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.

a(n) = [x^n y^9] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 21 2021

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, 10}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 9}];
Table[a[n], {n, 36, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz in A344257 *)

Crossrefs

Cf. A001222 (Omega), A001358.

Column k=9 of A344447.

Sequence in context: A344246 A344254 A344255 * A344257 A088931 A088980

Adjacent sequences: A344253 A344254 A344255 * A344257 A344258 A344259

Keyword

nonn

Author

Wesley Ivan Hurt, May 13 2021

Extensions

More terms from Alois P. Heinz, May 18 2021

Status

approved