A308843 Sum of the third largest parts in the partitions of n into 5 squarefree parts.

0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 12, 13, 21, 23, 32, 33, 49, 56, 77, 86, 117, 130, 162, 174, 223, 239, 295, 312, 391, 418, 497, 520, 631, 675, 801, 844, 1009, 1072, 1247, 1306, 1537, 1628, 1890, 1972, 2312, 2425, 2786, 2889, 3325, 3472, 3955, 4089, 4671, 4851, 5474

Offset: 0

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

• nonn

Comments

Conjecture: a(4*k + 3) < a(4*k + 4) for 4*k + 3 >= 195. This conjecture holds for all terms in the b-file. - David A. Corneth, Sep 16 2019

Links

• David A. Corneth, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A308839, A308840, A308841, A308842, A308844, A308845.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[j * MoebiusMu[l]^2*MoebiusMu[k]^2*MoebiusMu[j]^2 *MoebiusMu[i]^2*MoebiusMu[n - i - j - k - l]^2, {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{5}],AllTrue[#,SquareFreeQ]&][[All,3]]],{n,0,60}] (* Harvey P. Dale, Dec 26 2022 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * j, where mu is the Möbius function (A008683).

a(n) = A308839(n) - A308841(n) - A308842(n) - A308844(n) - A308845(n).

Extensions

a(54)..a(55) from David A. Corneth, Sep 16 2019