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

Author

Wesley Ivan Hurt, Dec 29 2016

Keywords

• nonn
• easy

Examples

			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)
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

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

Programs

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 *)

PARI

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

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