A326687 Sum of the second largest parts in 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, 3, 5, 6, 8, 14, 14, 19, 26, 34, 38, 49, 53, 70, 81, 96, 105, 139, 140, 187, 204, 246, 263, 326, 341, 437, 452, 543, 562, 715, 700, 898, 904, 1100, 1101, 1387, 1343, 1720, 1643, 2037, 1982

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, A326680, A326681, A326682, A326683, A326684, A326685, A326686, A326688.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,PrimeQ]&][[All,2]]],{n,0,70}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 04 2021 *)

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) * i, where c is the prime characteristic (A010051).

a(n) = A326678(n) - A326679(n) - A326680(n) - A326681(n) - A326682(n) - A326683(n) - A326684(n) - A326685(n) - A326686(n) - A326688(n).