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%I #9 Oct 15 2021 15:01:55
%S 0,0,0,0,0,0,0,0,0,0,0,0,12,13,14,30,32,51,72,57,80,105,132,138,192,
%T 175,260,270,336,319,480,372,608,561,748,630,936,740,1178,936,1320,
%U 1107,1764,1247,2068,1575,2346,1786,2880,2009,3400,2397,3796,2809,4644
%N Sum of all the parts in the partitions of n into 6 primes.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = n * Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-k-j-l-m), where c = A010051.
%F a(n) = n * A259196(n).
%F a(n) = A308920(n) + A308921(n) + A308922(n) + A308923(n) + A308924(n) + A308925(n).
%t Table[n*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[n - i - j - k - l - m] - PrimePi[n - i - j - k - l - m - 1]), {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 50}]
%Y Cf. A010051, A259196, A308920, A308921, A308922, A308923, A308924, A308925.
%K nonn
%O 0,13
%A _Wesley Ivan Hurt_, Jun 30 2019
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