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a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} (A010051c(q) + A010051c(p) + A010051c(o) + A010051c(m) + A010051c(l) + A010051c(k) + A010051c(j) + A010051c(i) + A010051c(n-i-j-k-l-m-o-p-q)), where c = A010051.

Table[Sum[Sum[Sum[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[o] - PrimePi[o - 1]) + (PrimePi[p] - PrimePi[p - 1]) + (PrimePi[q] - PrimePi[q - 1]) + (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

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Table[Count[Flatten[IntegerPartitions[n, {9}]], _?PrimeQ], {n, 0, 50}] (* Harvey P. Dale, Jun 12 2021 *)

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allocated for Wesley Ivan HurtNumber of prime parts in the partitions of n into 9 parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 5, 11, 17, 30, 45, 72, 104, 147, 200, 279, 367, 491, 633, 825, 1042, 1330, 1649, 2063, 2531, 3116, 3776, 4597, 5510, 6627, 7878, 9381, 11058, 13059, 15275, 17895, 20802, 24191, 27942, 32303, 37099, 42628, 48719, 55678

0,12

<a href="/index/Par#part">Index entries for sequences related to partitions</a>

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} (A010051(q) + A010051(p) + A010051(o) + A010051(m) + A010051(l) + A010051(k) + A010051(j) + A010051(i) + A010051(n-i-j-k-l-m-o-p-q)).

Table[Sum[Sum[Sum[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[o] - PrimePi[o - 1]) + (PrimePi[p] - PrimePi[p - 1]) + (PrimePi[q] - PrimePi[q - 1]) + (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Cf. A010051, A259200.

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Wesley Ivan Hurt, Aug 03 2019

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