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Let $\mathcal{A}$ be the set of all integers of the form $\gcd(n, F_n)$,where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number.We prove that $\#\left(\mathcal{A} \cap [1, x]\right) \gg x / \log x$ for all$x \geq 2$, and that $\mathcal{A}$ has zero asymptotic density. Our proofs relyon 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 $n$ dividesthe rank of appearance of $p$, that is, the smallest positive integer $k$ suchthat $p$ divides $F_k$.

On the greatest common divisor of $n$ and the $n$th Fibonacci number