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Table[n*Sum[Sum[Sum[Sum[Sum[Sum[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]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

approved

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editing

approved

Table[Total[Flatten[Select[IntegerPartitions[n, {7}], AllTrue[#, SquareFreeQ]&]]], {n, 0, 50}] (* Harvey P. Dale, Feb 25 2024 *)

approved

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proposed

approved

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proposed

a(n) = A308954(n) + A308955(n) + A308956(n) + A308957(n) + A308958(n) + A308959(n) + A308960(n).

Cf. A008683, A308952, A308954, A308955, A308956, A308957, A308958, A308959, A308960.

allocated for Wesley Ivan HurtSum of all the parts in the partitions of n into 7 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 7, 8, 18, 20, 44, 60, 104, 126, 180, 224, 340, 396, 551, 640, 882, 1034, 1357, 1536, 2025, 2314, 2943, 3304, 4176, 4680, 5797, 6464, 7887, 8806, 10605, 11664, 14023, 15504, 18291, 20040, 23657, 25956, 30272, 32956, 38295, 41860, 48269

0,8

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

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

a(n) = n * A308952(n).

Table[n*Sum[Sum[Sum[Sum[Sum[Sum[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]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

Cf. A008683, A308952.

allocated

nonn

Wesley Ivan Hurt, Jul 03 2019

approved

editing

allocated for Wesley Ivan Hurt

recycled

allocated

recycled

approved