A308783 Sum of all the parts in the partitions of n into 4 squarefree parts.

0, 0, 0, 0, 4, 5, 12, 14, 32, 36, 60, 66, 96, 104, 154, 165, 240, 255, 342, 380, 500, 504, 660, 690, 888, 900, 1144, 1161, 1484, 1508, 1800, 1860, 2272, 2277, 2720, 2800, 3348, 3404, 4028, 4056, 4880, 4879, 5670, 5762, 6820, 6840, 7912, 8084, 9312, 9408

Offset: 0

Author

Wesley Ivan Hurt, Jun 24 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A308767, A308762, A308768, A308769, A308770.

Programs

Mathematica

Table[n*Sum[Sum[Sum[MoebiusMu[k]^2*MoebiusMu[j]^2*MoebiusMu[i]^2* MoebiusMu[n - i - j - k]^2, {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[#,SquareFreeQ]&]]],{n,0,50}] (* Harvey P. Dale, Aug 14 2022 *)

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)} mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k)^2, where mu is the Möbius function (A008683).

a(n) = n * A308767(n).

a(n) = A308768(n) + A308762(n) + A308769(n) + A308770(n).