OFFSET
1,6
COMMENTS
a(p^k) = mu(p-1)^2 for k = 1 or 2, and 0 for k > 2.
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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Apr 24 2020
STATUS
approved