On the greatest common divisor ofnand thenth Fibonacci number, II

Abstract

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 that𝑥 log log log 𝑥 log log 𝑥 for all sufficiently large 𝑥. In fact, we prove that a similar bound holds also when the sequence of Fibonacci numbers is replaced by a general nondegenerate Lucas sequence.