On the greatest common divisor of n and the nth Fibonacci number, II

Let A be the set of all integers of the form gcd(n, F-n), where n is a positive integer and F-n denotes the nth Fibonacci number. Leonetti and Sanna proved that A has natural density equal to zero, and asked for a more precise upper bound. We prove that#(A boolean AND [1, x] << x log log log x/log log xfor all sufficiently large x. In fact, we prove that a similar bound also holds when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.