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A330868 Number of proper divisors d of n such that n-d is squarefree.

%I A330868 #22 May 03 2020 03:11:17
%S A330868 0,1,1,2,0,2,1,2,1,1,1,3,0,2,2,2,0,2,0,3,1,2,1,3,0,1,1,3,0,2,1,2,2,2,
%T A330868 2,4,0,2,2,4,0,4,1,4,2,1,1,3,1,0,1,3,0,2,0,2,1,2,1,5,0,2,2,1,0,3,1,4,
%U A330868 2,3,1,4,0,2,2,3,2,3,1,3,1,1,1,6,0,2,2,4,0,3
%N A330868 Number of proper divisors d of n such that n-d is squarefree.
%F A330868 a(n) = Sum_{d|n, d<n} mu(n-d)^2, where mu is the Möbius function (A008683).
%F A330868 Let m = p^k, where p is a prime and k is a positive integer. Then a(p^k) = Sum_{i=0..k-1} mu(p^k - p^i)^2. In terms of m, a(m) = Sum_{j=0..Omega(m)-1} mu(m - rad(m)^j)^2, where mu = A008683, Omega = A001222 and rad = A001222.
%e A330868 a(11) = 1; The only proper divisor of 11 is 1 and 11-1 = 10 is squarefree.
%e A330868 a(12) = 3; There are five proper divisors of 12: 1, 2, 3, 4, 6. Of these, we see that 12-1 = 11, 12-2 = 10 and 12-6 = 6 are squarefree, but 12-3 = 9 and 12-4 = 8 are not.
%e A330868 a(13) = 0; The only proper divisor of 13 is 1, but 13-1 = 12 (which is not squarefree).
%e A330868 a(14) = 2; The proper divisors of 14 are 1, 2, and 7. Of these, only 14-1 = 13 and 14-7 = 7 are squarefree.
%t A330868 Table[Sum[MoebiusMu[n - i]^2*(1 - Ceiling[n/i] + Floor[n/i]), {i, n - 1}], {n, 100}]
%o A330868 (PARI) a(n) = sumdiv(n, d, (d<n) && issquarefree(n-d)); \\ _Michel Marcus_, Apr 29 2020
%Y A330868 Cf. A001222 (Omega), A007947 (rad), A008683 (Möbius).
%Y A330868 Cf. A293227.
%K A330868 nonn,easy
%O A330868 1,4
%A A330868 _Wesley Ivan Hurt_, Apr 28 2020