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Number of proper divisors of n such that d, n/d and n-d are all squarefree.
1

%I #18 Apr 27 2020 00:24:48

%S 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,

%T 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,

%U 2,3,1,0,0,2,1,2,2,3,1,0,0,1,1,4,0,2,2,0,0,1,1,1

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

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

%F 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).

%e a(41) = 0; There are no such divisors of 41 since 1 and 41 are squarefree, but 41 - 1 = 40 is not.

%e a(42) = 4; The four divisors of 42 that meet all three conditions are 1, 3, 7 and 21.

%e a(43) = 1; The only divisor of 43 that meets all three conditions is 1.

%e a(44) = 2; The two divisors of 44 that meet all three conditions are 2 and 22.

%t 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}]

%o (PARI) a(n) = sumdiv(n, d, issquarefree(d) && issquarefree(n-d) && issquarefree(n/d)); \\ _Michel Marcus_, Apr 25 2020

%Y Cf. A000005, A008683, A332696.

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

%K nonn,easy

%O 1,6

%A _Wesley Ivan Hurt_, Apr 24 2020