A326683 Sum of the sixth 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, 2, 4, 4, 6, 9, 9, 12, 16, 20, 21, 27, 26, 36, 38, 47, 49, 68, 62, 86, 88, 111, 107, 145, 133, 183, 173, 220, 206, 287, 247, 341, 311, 411, 372, 509, 438, 610, 527, 713, 624, 865, 716, 1009

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, A326684, A326685, A326686, A326687, A326688.

Programs

Mathematica

Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,PrimeQ]&][[All,6]]],{n,0,70}] (* Harvey P. Dale, Aug 12 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) * m, where c = A010051.

a(n) = A326678(n) - A326679(n) - A326680(n) - A326681(n) - A326682(n) - A326684(n) - A326685(n) - A326686(n) - A326687(n) - A326688(n).