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

1,5

Links

Table of n, a(n) for n=1..73.

Index entries for sequences related to partitions

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

Cf. A008683, A008966, A294097.

Sequence in context: A240698 A010602 A120731 * A087477 A029212 A035687

Adjacent sequences: A294095 A294096 A294097 * A294099 A294100 A294101

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 22 2017

Status

approved