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A308923
Sum of the third largest parts in the partitions of n into 6 primes.
6
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 5, 8, 10, 9, 11, 16, 18, 21, 28, 27, 36, 41, 48, 46, 67, 54, 82, 78, 99, 86, 126, 104, 156, 129, 181, 152, 238, 175, 277, 221, 325, 249, 405, 295, 480, 342, 542, 394, 660, 430, 752, 517, 851, 584, 1005, 643
OFFSET
0,13
FORMULA
a(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-j-k-l-m) * j, where c = A010051.
a(n) = A308919(n) - A308920(n) - A308921(n) - A308922(n) - A308924(n) - A308925(n).
MATHEMATICA
Table[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[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}]
Table[Total[Select[IntegerPartitions[n, {6}], AllTrue[#, PrimeQ]&][[;; , 3]]], {n, -0, 70}] (* Harvey P. Dale, Sep 24 2024 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jun 30 2019
STATUS
approved