A351207 Number of ordered pairs of divisors of n whose sum is squarefree.

1, 3, 2, 7, 4, 8, 2, 9, 4, 12, 2, 22, 4, 8, 6, 11, 2, 18, 2, 24, 8, 10, 2, 32, 8, 10, 6, 22, 4, 32, 2, 15, 8, 10, 8, 48, 4, 10, 8, 32, 4, 28, 2, 24, 12, 8, 2, 40, 2, 22, 4, 24, 2, 28, 8, 34, 8, 12, 2, 80, 4, 8, 16, 19, 14, 34, 2, 26, 8, 32, 2, 72, 4, 10, 12, 26, 6, 30, 2, 42

Offset

1,2

Links

Table of n, a(n) for n=1..80.

Formula

a(n) = Sum_{d1|n, d2|n} mu(d1 + d2)^2.

Example

a(6) = 8; There are 8 ordered pairs of divisors of 6 whose sum is squarefree: (1,1), (1,2), (1,6), (2,1), (2,3), (3,2), (3,3), (6,1).

Mathematica

a[n_] := Count[Tuples[Divisors[n], 2], _?(SquareFreeQ[Total[#]] &)]; Array[a, 80] (* Amiram Eldar, Feb 05 2022 *)

Prog

(PARI) a(n) = my(d=divisors(n)); sum (i=1, #d, sum (j=1, #d, issquarefree(d[i]+d[j]))); \\ Michel Marcus, Feb 05 2022

Crossrefs

Cf. A008683 (mu), A008966 (squarefree).

Sequence in context: A178910 A182651 A175055 * A085168 A327119 A113658

Adjacent sequences: A351204 A351205 A351206 * A351208 A351209 A351210

Keyword

nonn

Author

Wesley Ivan Hurt, Feb 04 2022

Status

approved