A340227 - OEIS

%I #6 May 21 2021 11:51:21

%S 0,1,1,1,1,4,1,1,1,4,1,4,1,4,4,1,1,4,1,4,4,4,1,4,1,4,1,4,1,13,1,1,4,4,

%T 4,4,1,4,4,4,1,13,1,4,4,4,1,4,1,4,4,4,1,4,4,4,4,4,1,13,1,4,4,1,4,13,1,

%U 4,4,13,1,4,1,4,4,4,4,13,1,4,1,4,1,13,4,4,4,4,1,13

%N Number of pairs of divisors of n, (d1,d2), such that d1 < d2 and d1*d2 is squarefree.

%C If n = p where p is prime, the only pair of divisors of n such that d1 < d2 is (1,p). Since the product 1*p = p is squarefree, this satisfies the constraints. Thus, a(p) = 1 for all p. - _Wesley Ivan Hurt_, May 21 2021

%F Sum_{d1|n, d2|n, d1<d2} mu(d1*d2)^2, where mu is the Möbius function (A008683).

%e a(28) = 4; (1,2), (1,7), (1,14), (2,7)

%e a(29) = 1; (1,29)

%e a(30) = 13; (1,2), (1,3), (1,5), (1,6), (1,10), (1,15), (1,30), (2,3), (2,5), (2,15), (3,5), (3,10), (5,6)

%e a(31) = 1; (1,31)

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

%Y Cf. A008683.

%K nonn

%O 1,6

%A _Wesley Ivan Hurt_, Jan 01 2021