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A326547 Sum of the third largest parts in the partitions of n into 9 primes.

%I A326547 #9 Aug 31 2025 15:26:09
%S A326547 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,2,5,5,8,10,12,14,22,21,30,34,
%T A326547 41,44,65,64,83,88,108,115,154,142,189,191,241,233,314,289,384,369,
%U A326547 472,455,604,537,729,687,889,816,1089,974,1318,1179,1548,1392
%N A326547 Sum of the third largest parts in the partitions of n into 9 primes.
%H A326547 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A326547 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)} 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) * j, where c = A010051.
%F A326547 a(n) = A326540(n) - A326541(n) - A326542(n) - A326543(n) - A326544(n) - A326545(n) - A326546(n) - A326548(n) - A326549(n).
%t A326547 Table[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[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}]
%Y A326547 Cf. A010051, A259200, A326540, A326541, A326542, A326543, A326544, A326545, A326546, A326548, A326549.
%K A326547 nonn,changed
%O A326547 0,19
%A A326547 _Wesley Ivan Hurt_, Jul 13 2019