A308770 Sum of the largest parts of the partitions of n into 4 squarefree parts.

0, 0, 0, 0, 1, 2, 5, 5, 13, 17, 29, 32, 44, 52, 75, 81, 118, 130, 176, 198, 261, 262, 351, 362, 470, 478, 617, 621, 787, 801, 951, 978, 1182, 1184, 1413, 1469, 1747, 1789, 2123, 2160, 2574, 2593, 3012, 3093, 3644, 3679, 4245, 4384, 5024, 5097, 5738, 5891

Offset

0,6

Links

Table of n, a(n) for n=0..51.

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

a(n) = A308783(n) - A308768(n) - A308762(n) - A308769(n).

Mathematica

Table[Sum[Sum[Sum[(n - i - j - k) * 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, 100}]
Table[Total[Select[IntegerPartitions[n, {4}], AllTrue[#, SquareFreeQ]&][[;; , 1]]], {n, 0, 60}] (* Harvey P. Dale, Oct 04 2023 *)

Crossrefs

Cf. A008683, A308762, A308767, A308768, A308769, A308783.

Sequence in context: A112835 A206625 A176168 * A222114 A308845 A308911

Adjacent sequences: A308767 A308768 A308769 * A308771 A308772 A308773

Keyword

nonn

Author

Wesley Ivan Hurt, Jun 23 2019

Status

approved