A326461 Sum of the third largest parts in the partitions of n into 8 primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 5, 8, 10, 12, 14, 19, 21, 27, 34, 35, 44, 57, 64, 67, 88, 87, 115, 121, 142, 146, 191, 176, 233, 232, 289, 271, 369, 336, 455, 414, 537, 500, 687, 588, 816, 722, 974, 843, 1179, 977, 1392, 1172

Offset: 0

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326460, A326462, A326463.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{8}],AllTrue[#,PrimeQ]&][[;;,3]]],{n,0,70}] (* Harvey P. Dale, Apr 13 2025 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * j, where c = A010051.

a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326462(n) - A326463(n).