On the greatest common divisor of n and the 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/logx for all x≥2 and that A has zero asymptotic density. Our proofs rely upon a recent result of Cubre and Rouse 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.