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

40,7

Links

Alois P. Heinz, Table of n, a(n) for n = 40..10000

Index entries for sequences related to partitions

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

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

Crossrefs

Cf. A001222 (Omega), A001358.

Column k=10 of A344447.

Sequence in context: A344254 A344255 A344256 * A088931 A088980 A254436

Adjacent sequences: A344254 A344255 A344256 * A344258 A344259 A344260

Keyword

nonn

Author

Wesley Ivan Hurt, May 13 2021

Extensions

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

Status

approved