A326628 - OEIS

%I #12 Sep 16 2024 19:27:42

%S 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,

%T 124,142,176,199,248,280,338,382,461,519,615,688,819,914,1066,1186,

%U 1392,1539,1780,1970,2278,2512,2879,3167,3629,3986,4517,4960,5631,6159

%N Sum of the smallest parts of the partitions of n into 10 squarefree parts.

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

%F 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).

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

%t 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}]

%t Table[Total[Select[IntegerPartitions[n,{10}],AllTrue[#,SquareFreeQ]&][[;;,-1]]],{n,0,60}] (* _Harvey P. Dale_, Sep 16 2024 *)

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

%K nonn

%O 0,13

%A _Wesley Ivan Hurt_, Jul 14 2019