A308770 - OEIS

%I #9 Oct 04 2023 11:11:15

%S 0,0,0,0,1,2,5,5,13,17,29,32,44,52,75,81,118,130,176,198,261,262,351,

%T 362,470,478,617,621,787,801,951,978,1182,1184,1413,1469,1747,1789,

%U 2123,2160,2574,2593,3012,3093,3644,3679,4245,4384,5024,5097,5738,5891

%N Sum of the largest parts of the partitions of n into 4 squarefree parts.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F 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).

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

%t 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}]

%t Table[Total[Select[IntegerPartitions[n,{4}],AllTrue[#,SquareFreeQ]&][[;;,1]]],{n,0,60}] (* _Harvey P. Dale_, Oct 04 2023 *)

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

%K nonn

%O 0,6

%A _Wesley Ivan Hurt_, Jun 23 2019