On the greatest common divisor of n and nth Fibonacci number

Let A be the set of all integers of the form gcd(n; Fn), where n is a positive integer and Fn denotes the nth Fibonacci number. We prove that #(A ∩ [1; x]) ≫ x= log x for all x ≥ 2 and that A has zero asymptotic density. Our proofs rely upon a recent result of Cubre and Rouse [5] which gives, for each positive integer n, an explicit formula for the density of primes p such that n divides the rank of appearance of p, that is, the smallest positive integer k such that p divides Fk.