A280320 Sum of the squares of the smaller parts of the partitions of 2n into two squarefree parts.

1, 5, 10, 14, 34, 66, 59, 75, 84, 220, 205, 309, 373, 600, 565, 665, 839, 1103, 959, 1191, 1176, 1860, 1416, 2060, 1664, 3653, 2194, 3505, 2891, 4974, 3563, 5534, 4371, 7551, 5845, 8874, 6742, 10409, 7061, 10145, 8037, 12414, 9030, 13327, 10849, 15319, 13473, 15960

Offset: 1

Author

Wesley Ivan Hurt, Dec 31 2016

Keywords

• nonn
• easy

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A280226, A280316, A280322.

Programs

Maple

with(numtheory): A280320:=n->add(i^2*mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280320(n), n=1..100);

Mathematica

Table[Total[Select[IntegerPartitions[2 n,{2}],AllTrue[#,SquareFreeQ]&][[All,2]]^2],{n,50}] (* Harvey P. Dale, Jan 22 2023 *)

PARI

a(n) = sum(i=1, n, i^2*issquarefree(i)*issquarefree(2*n-i)); \\ Michel Marcus, May 16 2019

Formula

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

a(n) = A280316(n) - A280322(n).