A308919 Sum of all the parts in the partitions of n into 6 primes.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 13, 14, 30, 32, 51, 72, 57, 80, 105, 132, 138, 192, 175, 260, 270, 336, 319, 480, 372, 608, 561, 748, 630, 936, 740, 1178, 936, 1320, 1107, 1764, 1247, 2068, 1575, 2346, 1786, 2880, 2009, 3400, 2397, 3796, 2809, 4644

Offset: 0

Author

Wesley Ivan Hurt, Jun 30 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A259196, A308920, A308921, A308922, A308923, A308924, A308925.

Programs

Mathematica

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

Formula

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

a(n) = n * A259196(n).

a(n) = A308920(n) + A308921(n) + A308922(n) + A308923(n) + A308924(n) + A308925(n).