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

0,5

Links

Table of n, a(n) for n=0..49.

Index entries for sequences related to partitions

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).

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 *)

Crossrefs

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

Sequence in context: A080277 A047608 A266725 * A130011 A050022 A137619

Adjacent sequences: A308780 A308781 A308782 * A308784 A308785 A308786

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 24 2019

Status

approved