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A294098
Number of partitions of 2n into two parts such that one part is squarefree and the other part is nonsquarefree.
1
0, 0, 1, 0, 3, 1, 4, 1, 4, 2, 5, 2, 7, 4, 9, 4, 8, 1, 11, 4, 12, 4, 14, 5, 15, 5, 13, 8, 14, 8, 17, 9, 19, 7, 18, 3, 19, 8, 23, 10, 25, 9, 26, 9, 22, 12, 25, 12, 27, 11, 27, 12, 28, 5, 31, 12, 32, 12, 34, 13, 36, 12, 31, 18, 34, 18, 37, 19, 39, 17, 40, 7, 41
OFFSET
1,5
FORMULA
a(n) = n - Sum_{i=1..n} [c(i) = c(2*n-i)], where [] is the Iverson bracket and c is the squarefree characteristic (A008966).
a(n) = Sum_{i=1..n} mu(i)^2 * (1-mu(2*n-i)^2) + (1-mu(i)^2) * mu(2*n-i)^2, where mu is the Möbius function (A008683). - Wesley Ivan Hurt, Nov 18 2017
MATHEMATICA
Table[n - Sum[KroneckerDelta[MoebiusMu[k]^2, MoebiusMu[2 n - k]^2], {k, n}], {n, 80}]
Table[Count[IntegerPartitions[2n, {2}], _?(Total[Boole[ SquareFreeQ/@#]] == 1&)], {n, 80}] (* Harvey P. Dale, Jul 27 2021 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 22 2017
STATUS
approved