A280226 Number of partitions of 2n into two squarefree parts.

1, 2, 2, 3, 2, 4, 3, 5, 4, 6, 5, 7, 5, 7, 5, 8, 7, 11, 7, 11, 8, 13, 8, 13, 8, 14, 10, 13, 11, 15, 11, 15, 11, 18, 13, 21, 14, 20, 13, 20, 13, 22, 14, 23, 17, 23, 17, 24, 17, 25, 18, 26, 19, 31, 19, 29, 20, 31, 20, 31, 20, 33, 23, 30, 23, 32, 23, 32, 23, 35, 24, 41, 25, 39

Offset

1,2

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

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

a(n) = n - A302391(n). - Wesley Ivan Hurt, Dec 11 2023

Example

From Wesley Ivan Hurt, Feb 20 2018: (Start)
a(5) = 2; there are two partitions of 2*5 = 10 into two squarefree parts: (7,3), (5,5).
a(6) = 4; there are four partitions of 2*6 = 12 into two squarefree parts: (11,1), (10,2), (7,5), (6,6).
a(7) = 3; there are three partitions of 2*7 = 14 into two squarefree parts: (13,1), (11,3), (7,7).
a(8) = 5; there are five partitions of 2*8 = 16 into two squarefree parts: (15,1), (14,2), (13,3), (11,5), (10,6). (End)

Maple

with(numtheory): A280226:=n->sum(mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280226(n), n=1..100);

Mathematica

f[n_] := Sum[(MoebiusMu[i]*MoebiusMu[2n -i])^2, {i, n}]; Array[f, 74] (* Robert G. Wilson v, Dec 29 2016 *)

Prog

(PARI) a(n)=sum(i=1, n, issquarefree(i) && issquarefree(2*n-i)) \\ Charles R Greathouse IV, Nov 05 2017

Crossrefs

Cf. A008683, A045917, A262991, A280250, A280251, A280252, A294248, A302391.

Sequence in context: A322355 A242112 A211316 * A307995 A061889 A240089

Adjacent sequences: A280223 A280224 A280225 * A280227 A280228 A280229

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 29 2016

Status

approved