A326445 - OEIS

%I #11 Jul 21 2024 10:49:39

%S 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,

%T 122,153,169,207,231,280,310,371,409,487,535,630,688,812,883,1028,

%U 1119,1295,1409,1619,1754,2014,2180,2479,2679,3046,3284,3707,3994,4502

%N Sum of the smallest parts of the partitions of n into 8 squarefree parts.

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

%F 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).

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

%t 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}]

%t Table[Total[Select[IntegerPartitions[n,{8}],AllTrue[#,SquareFreeQ]&][[;;,-1]]],{n,0,60}] (* _Harvey P. Dale_, Jul 21 2024 *)

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

%K nonn

%O 0,11

%A _Wesley Ivan Hurt_, Jul 06 2019