A308975 Sum of the smallest parts of the partitions of n into 7 primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 9, 8, 13, 12, 18, 16, 20, 20, 32, 24, 37, 32, 45, 38, 63, 44, 74, 52, 84, 62, 109, 66, 123, 84, 145, 94, 173, 102, 209, 120, 225, 136, 272, 146, 309, 172, 343, 190, 405, 206, 466, 232, 499, 262

Offset: 0

Author

Wesley Ivan Hurt, Jul 04 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A259197, A307637, A308974, A308976, A308977, A308978, A308979, A308980.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[o*(PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Formula

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

a(n) = A308974(n) - A308976(n) - A308977(n) - A308978(n) - A308979(n) - A307637(n) - A308980(n).