A294233 Number of partitions of n into two parts with smaller part nonsquarefree and larger part squarefree.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 2, 3, 1, 2, 2, 3, 1, 3, 3, 3, 1, 2, 3, 3, 1, 3, 3, 4, 0, 4, 5, 5, 2, 5, 6, 5, 2, 4, 6, 6, 2, 5, 7, 8, 2, 5, 6, 9, 4, 7, 7, 9, 4, 7, 8, 8, 4, 8, 9, 9, 3, 8, 8, 10, 1, 9, 9, 10, 3, 8, 10, 9, 4, 8, 11, 11, 3, 9

Offset: 1

Author

Wesley Ivan Hurt, Oct 25 2017

Keywords

• nonn
• easy

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A008966, A294101, A294232.

Programs

Mathematica

Table[Sum[(1 - MoebiusMu[k]^2) MoebiusMu[n - k]^2, {k, Floor[n/2]}], {n, 80}]

Formula

a(n) = Sum_{i=1..floor(n/2)} mu(n-i)^2 * (1 - mu(i)^2), where mu is the Möbius function (A008683).

a(n) = A294101(n) - A294232(n). - Wesley Ivan Hurt, Jul 16 2025