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

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

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

Programs

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 *)

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).