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

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

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

Programs

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 *)

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).