A326628 Sum of the smallest parts of the partitions of n into 10 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 16, 24, 27, 36, 43, 57, 66, 84, 96, 124, 142, 176, 199, 248, 280, 338, 382, 461, 519, 615, 688, 819, 914, 1066, 1186, 1392, 1539, 1780, 1970, 2278, 2512, 2879, 3167, 3629, 3986, 4517, 4960, 5631, 6159

Offset

0,13

Links

Table of n, a(n) for n=0..57.

Index entries for sequences related to partitions

Formula

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} mu(r)^2 * mu(q)^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-q-r)^2 * r, where mu is the Möbius function (A008683).

a(n) = A326627(n) - A326629(n) - A326630(n) - A326631(n) - A326632(n) - A326633(n) - A326634(n) - A326635(n) - A326636(n) - A326637(n).

Mathematica

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[r * MoebiusMu[r]^2 * MoebiusMu[q]^2 * 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 - q - r]^2 , {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n, {10}], AllTrue[#, SquareFreeQ]&][[;; , -1]]], {n, 0, 60}] (* Harvey P. Dale, Sep 16 2024 *)

Crossrefs

Cf. A008683, A326626, A326627, A326629, A326630, A326631, A326632, A326633, A326634, A326635, A326636, A326637.

Sequence in context: A326522 A326626 A341153 * A326524 A326629 A326445

Adjacent sequences: A326625 A326626 A326627 * A326629 A326630 A326631

Keyword

nonn

Author

Wesley Ivan Hurt, Jul 14 2019

Status

approved