A326684 - OEIS

%I #7 Sep 17 2021 18:05:08

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,4,4,7,9,10,12,18,20,23,

%T 27,30,36,45,49,59,72,75,92,106,119,130,157,165,201,210,244,257,319,

%U 307,383,386,465,463,575,547,694,654,815,784,991,896,1163

%N Sum of the fifth largest parts of the partitions of n into 10 primes.

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

%F 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) * l, where c = A010051.

%F a(n) = A326678(n) - A326679(n) - A326680(n) - A326681(n) - A326682(n) - A326683(n) - A326685(n) - A326686(n) - A326687(n) - A326688(n).

%t Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[l * (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}]

%Y Cf. A010051, A259201, A326678, A326679, A326680, A326681, A326682, A326683, A326685, A326686, A326687, A326688.

%K nonn

%O 0,21

%A _Wesley Ivan Hurt_, Jul 17 2019