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A326524
Sum of the smallest parts of the partitions of n into 9 squarefree parts.
10
0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 9, 13, 17, 23, 26, 36, 42, 55, 63, 80, 93, 119, 131, 165, 188, 230, 255, 312, 351, 420, 466, 555, 620, 731, 804, 945, 1046, 1216, 1333, 1550, 1702, 1959, 2141, 2452, 2688, 3064, 3334, 3790, 4136, 4673, 5070
OFFSET
0,12
FORMULA
a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/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)^2 * q, where mu is the Möbius function (A008683).
a(n) = A326523(n) - A326525(n) - A326526(n) - A326527(n) - A326528(n) - A326529(n) - A326530(n) - A326531(n) - A326532(n).
MATHEMATICA
Table[Total[Select[IntegerPartitions[n, {9}], AllTrue[#, SquareFreeQ]&][[;; , -1]]], {n, 0, 60}] (* Harvey P. Dale, Mar 22 2023 *)
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Jul 11 2019
STATUS
approved