A326680 Sum of the ninth largest parts of the partitions of n into 10 primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 8, 10, 15, 17, 19, 23, 24, 30, 34, 38, 44, 54, 53, 67, 73, 83, 87, 105, 107, 131, 136, 156, 161, 200, 186, 233, 232, 275, 271, 335, 315, 398, 373, 456, 439, 550, 493, 636, 589

Offset: 0

Author

Wesley Ivan Hurt, Jul 17 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A259201, A326678, A326679, A326681, A326682, A326683, A326684, A326685, A326686, A326687, A326688.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,PrimeQ]&][[All,9]]],{n,0,70}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 01 2019 *)

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)} c(r) * c(q) * c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p-q-r) * q, where c = A010051.

a(n) = A326678(n) - A326679(n) - A326681(n) - A326682(n) - A326683(n) - A326684(n) - A326685(n) - A326686(n) - A326687(n) - A326688(n).