A344257 - OEIS

%I #18 May 26 2021 05:21:38

%S 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,

%T 33,38,40,50,52,59,63,71,75,86,94,105,110,124,131,150,159,174,189,205,

%U 217,242,264,288,303,327,354,388,414,443,476,511,547,594,641

%N Number of partitions of n into 10 semiprime parts.

%H Alois P. Heinz, <a href="/A344257/b344257.txt">Table of n, a(n) for n = 40..10000</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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.

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

%p h:= proc(n) option remember; `if`(n=0, 0,

%p      `if`(numtheory[bigomega](n)=2, n, h(n-1)))

%p     end:

%p b:= proc(n, i) option remember; series(`if`(n=0, 1, `if`(i<1, 0,

%p      `if`(i>n, 0, x*b(n-i, h(min(n-i, i))))+b(n, h(i-1)))), x, 11)

%p     end:

%p a:= n-> coeff(b(n, h(n)), x, 10):

%p seq(a(n), n=40..120);  # _Alois P. Heinz_, May 26 2021

%Y Cf. A001222 (Omega), A001358.

%Y Column k=10 of A344447.

%K nonn

%O 40,7

%A _Wesley Ivan Hurt_, May 13 2021

%E a(83)-a(103) from _Alois P. Heinz_, May 18 2021