A326678 Sum of all the 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, 20, 21, 22, 46, 48, 75, 104, 108, 140, 203, 240, 279, 352, 363, 476, 560, 648, 740, 950, 936, 1240, 1353, 1596, 1677, 2112, 2115, 2714, 2773, 3312, 3381, 4350, 4080, 5304, 5194, 6372, 6270, 8008

Offset: 0

Author

Wesley Ivan Hurt, Jul 17 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A259201, A326679, A326680, A326681, A326682, A326683, A326684, A326685, A326686, A326687, A326688.

Programs

Mathematica

Table[Total[Flatten[Select[IntegerPartitions[n,{10}],AllTrue[#,PrimeQ]&]]],{n,0,60}] (* Harvey P. Dale, Jan 31 2024 *)

Formula

a(n) = 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), where c = A010051.

a(n) = n * A259201(n).

a(n) = A326679(n) + A326680(n) + A326681(n) + A326682(n) + A326683(n) + A326684(n) + A326685(n) + A326686(n) + A326687(n) + A326688(n).