A308773 Sum of the second largest parts in the partitions of n into 4 prime parts.

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 6, 5, 11, 8, 13, 12, 20, 17, 28, 15, 32, 26, 41, 24, 53, 33, 75, 48, 83, 57, 103, 54, 126, 80, 143, 71, 170, 93, 219, 112, 217, 122, 276, 120, 310, 145, 320, 148, 376, 160, 446, 190, 443, 218, 532, 196, 587, 240, 613, 246

Offset

0,9

Links

Table of n, a(n) for n=0..61.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} c(k) * c(j) * c(i) * c(n-i-j-k) * i, where c = A010051.

a(n) = A308809(n) - A308771(n) - A308772(n) - A308774(n).

Mathematica

Table[Sum[Sum[Sum[i (PrimePi[k] - PrimePi[k - 1])*(PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Table[Total[Select[IntegerPartitions[n, {4}], AllTrue[#, PrimeQ]&][[;; , 2]]], {n, 0, 70}] (* Harvey P. Dale, May 09 2025 *)

Crossrefs

Cf. A010051, A259194, A308771, A308772, A308774, A308809.

Sequence in context: A333468 A364971 A225489 * A051732 A098382 A098180

Adjacent sequences: A308770 A308771 A308772 * A308774 A308775 A308776

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 23 2019

Status

approved