A308809 Sum of all the parts in the partitions of n into 4 primes.

0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 10, 22, 24, 26, 42, 30, 48, 51, 72, 76, 120, 63, 132, 115, 168, 125, 234, 135, 308, 203, 330, 217, 416, 198, 476, 315, 540, 296, 684, 351, 840, 410, 798, 473, 1056, 450, 1196, 564, 1248, 637, 1500, 612, 1768, 795, 1782, 880

Offset: 0

Author

Wesley Ivan Hurt, Jun 25 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A010051, A259194, A308771, A308772, A308773, A308774.

Programs

Mathematica

Table[n*Sum[Sum[Sum[(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, 50}]
Table[Total[Flatten[Select[IntegerPartitions[n,{4}],AllTrue[#,PrimeQ]&]]],{n,0,60}] (* Harvey P. Dale, Sep 28 2024 *)

Formula

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

a(n) = n * A259194(n).

a(n) = A308771(n) + A308772(n) + A308773(n) + A308774(n).