The density of numbers n having a prescribed G.C.D. with the nth Fibonacci number

For each positive integer k, let Ak be the set of all positive integers n such that gcd(n,Fn)=k, where Fn denotes the nth Fibonacci number. We prove that the asymptotic density of Ak exists and is equal to ∑d=1∞μ(d)lcm(dk,z(dk))where μ is the Möbius function and z(m) denotes the least positive integer n such that m divides Fn. We also give an effective criterion to establish when the asymptotic density of Ak is zero and we show that this is the case if and only if Ak is empty.