A326445 Sum of the smallest parts of the partitions of n into 8 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 16, 22, 26, 35, 40, 52, 59, 78, 86, 108, 122, 153, 169, 207, 231, 280, 310, 371, 409, 487, 535, 630, 688, 812, 883, 1028, 1119, 1295, 1409, 1619, 1754, 2014, 2180, 2479, 2679, 3046, 3284, 3707, 3994, 4502

Offset: 0

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

• nonn

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A326443, A326444, A326446, A326447, A326448, A326449, A326450, A326451, A326452.

Programs

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[p * MoebiusMu[p]^2 * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o - p]^2, {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{8}],AllTrue[#,SquareFreeQ]&][[;;,-1]]],{n,0,60}] (* Harvey P. Dale, Jul 21 2024 *)

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} mu(p)^2 * mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o-p)^2 * p, where mu is the Möbius function (A008683).

a(n) = A326444(n) - A326446(n) - A326447(n) - A326448(n) - A326449(n) - A326450(n) - A326451(n) - A326452(n).