A294101 Number of partitions of n into two parts such that one is squarefree and the other is nonsquarefree.

0, 0, 0, 0, 1, 1, 1, 0, 2, 3, 3, 1, 2, 4, 4, 1, 3, 4, 6, 2, 5, 5, 7, 2, 6, 7, 7, 4, 7, 9, 9, 4, 8, 8, 10, 1, 9, 11, 11, 4, 10, 12, 10, 4, 9, 14, 14, 5, 11, 15, 17, 5, 12, 13, 19, 8, 14, 14, 18, 8, 15, 17, 17, 9, 17, 19, 19, 7, 18, 18, 22, 3, 19, 19, 21, 8

Offset: 1

Author

Wesley Ivan Hurt, Oct 22 2017

Keywords

• nonn
• easy

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683 (mu), A008966, A071068, A262868, A262869, A262991, A294100, A294232, A294233.

Programs

Mathematica

Table[Floor[n/2] - Sum[KroneckerDelta[MoebiusMu[k]^2, MoebiusMu[n - k]^2], {k, Floor[n/2]}], {n, 80}]

Formula

a(n) = floor(n/2) - Sum_{i=1..floor(n/2)} [mu(i)^2 = mu(n-i)^2], where [] is the Iverson bracket.

From Wesley Ivan Hurt, Jul 16 2025: (Start)

a(n) = A294232(n) + A294233(n).

a(n) = A262991(n) - 2*A071068(n). (End)