A294098 Number of partitions of 2n into two parts such that one part is squarefree and the other part is nonsquarefree.
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
Programs
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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 *)
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