A309471 Sum of the prime parts in the partitions of n into 10 parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 7, 11, 28, 41, 79, 115, 191, 273, 428, 574, 851, 1133, 1576, 2072, 2819, 3621, 4812, 6112, 7918, 9931, 12655, 15684, 19714, 24221, 29987, 36534, 44796, 54051, 65660, 78684, 94653, 112671, 134499, 159012, 188569, 221650

Offset: 0

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

• nonn

Links

• David A. Corneth, Table of n, a(n) for n = 0..9999
• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A309439, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + l (PrimePi[l] - PrimePi[l - 1]) + m (PrimePi[m] - PrimePi[m - 1]) + o (PrimePi[o] - PrimePi[o - 1]) + p (PrimePi[p] - PrimePi[p - 1]) + q (PrimePi[q] - PrimePi[q - 1]) + r (PrimePi[r] - PrimePi[r - 1]) + (n - i - j - k - l - m - o - p - q - r) (PrimePi[n - i - j - k - l - m - o - p - q - r] - PrimePi[n - i - j - k - l - m - o - p - q - r - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 80}]

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)} (r * c(r) + q * c(q) + p * c(p) + o * c(o) + m * c(m) + l * c(l) + k * c(k) + j * c(j) + i * c(i) + (n-i-j-k-l-m-o-p-q-r) * c(n-i-j-k-l-m-o-p-q-r)), where c is the prime characteristic (A010051).