A326686 Sum of the third largest parts of 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, 2, 2, 2, 5, 5, 8, 10, 12, 14, 22, 24, 30, 37, 41, 50, 65, 72, 83, 104, 108, 136, 154, 175, 189, 234, 241, 298, 314, 371, 384, 487, 472, 596, 604, 732, 729, 928, 889, 1134, 1089, 1360, 1318, 1685

Offset

0,21

Links

Table of n, a(n) for n=0..62.

Index entries for sequences related to partitions

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)} 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) * j, where c = A010051.

a(n) = A326678(n) - A326679(n) - A326680(n) - A326681(n) - A326682(n) - A326683(n) - A326684(n) - A326685(n) - A326687(n) - A326688(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[j * (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[r] - PrimePi[r - 1]) (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, 50}]

Crossrefs

Cf. A010051, A259201, A326678, A326679, A326680, A326681, A326682, A326683, A326684, A326685, A326687, A326688.

Sequence in context: A308979 A326461 A326547 * A096403 A073819 A190846

Adjacent sequences: A326683 A326684 A326685 * A326687 A326688 A326689

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 17 2019

Extensions

Name clarified by Sean A. Irvine, Jan 12 2025

Status

approved