A344257 Number of partitions of n into 10 semiprime parts.

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, 33, 38, 40, 50, 52, 59, 63, 71, 75, 86, 94, 105, 110, 124, 131, 150, 159, 174, 189, 205, 217, 242, 264, 288, 303, 327, 354, 388, 414, 443, 476, 511, 547, 594, 641

Offset: 40

Author

Wesley Ivan Hurt, May 13 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 40..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A001222 (Omega), A001358.

Column k=10 of A344447.

Programs

Maple

h:= proc(n) option remember; `if`(n=0, 0,
     `if`(numtheory[bigomega](n)=2, n, h(n-1)))
    end:
b:= proc(n, i) option remember; 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, 11)
    end:
a:= n-> coeff(b(n, h(n)), x, 10):
seq(a(n), n=40..120);  # Alois P. Heinz, May 26 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, 11}];
a[n_] := SeriesCoefficient[b[n, h[n]], {x, 0, 10}];
Table[a[n], {n, 40, 120}] (* Jean-François Alcover, May 15 2025, after Alois P. Heinz *)

Formula

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.

a(n) = [x^n y^10] 1/Product_{j>=1} (1-y*x^A001358(j)). - Alois P. Heinz, May 19 2021

Extensions

a(83)-a(103) from Alois P. Heinz, May 18 2021