40,7

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

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

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

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 *)

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

Wesley Ivan Hurt, May 13 2021

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

approved