A334368 Number of proper divisors of n such that d, n/d and n-d are all squarefree.

0, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 2, 0, 2, 2, 0, 0, 1, 0, 1, 1, 2, 1, 0, 0, 1, 0, 2, 0, 2, 1, 0, 2, 2, 2, 1, 0, 2, 2, 0, 0, 4, 1, 2, 2, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 1, 2, 1, 2, 0, 2, 1, 0, 0, 3, 1, 2, 2, 3, 1, 0, 0, 2, 1, 2, 2, 3, 1, 0, 0, 1, 1, 4, 0, 2, 2, 0, 0, 1, 1, 1

Offset

1,6

Comments

a(p^k) = mu(p-1)^2 for k = 1 or 2, and 0 for k > 2.

Links

Table of n, a(n) for n=1..92.

Formula

a(n) = Sum_{d|n, d<n} mu(d)^2 * mu(n/d)^2 * mu(n-d)^2, where mu is the Möebius function (A008683).

Example

a(41) = 0; There are no such divisors of 41 since 1 and 41 are squarefree, but 41 - 1 = 40 is not.
a(42) = 4; The four divisors of 42 that meet all three conditions are 1, 3, 7 and 21.
a(43) = 1; The only divisor of 43 that meets all three conditions is 1.
a(44) = 2; The two divisors of 44 that meet all three conditions are 2 and 22.

Mathematica

Table[Sum[MoebiusMu[i]^2 MoebiusMu[n/i]^2 MoebiusMu[n - i]^2 (1 - Ceiling[n/i] + Floor[n/i]), {i, Floor[n/2]}], {n, 100}]

Prog

(PARI) a(n) = sumdiv(n, d, issquarefree(d) && issquarefree(n-d) && issquarefree(n/d)); \\ Michel Marcus, Apr 25 2020

Crossrefs

Cf. A000005, A008683, A332696.

Cf. A007427 (with only d and n/d squarefree).

Sequence in context: A369460 A092876 A187360 * A240718 A357071 A292518

Adjacent sequences: A334365 A334366 A334367 * A334369 A334370 A334371

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 24 2020

Status

approved