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The density of numbers nn having a prescribed G.C.D. with the nnth Fibonacci number

Abstract

For each positive integer kk, let Ak\mathscr{A}_k be the set of all positive integers nn such that gcd(n,Fn)=k\gcd(n, F_n) = k, where FnF_n denotes the nnth Fibonacci number. We prove that the asymptotic density of Ak\mathscr{A}_k exists and is equal to d=1μ(d)lcm(dk,z(dk))\sum_{d = 1}^\infty \frac{\mu(d)}{\operatorname{lcm}(dk, z(dk))} where μ\mu is the M\"obius function and z(m)z(m) denotes the least positive integer nn such that mm divides FnF_n. We also give an effective criterion to establish when the asymptotic density of Ak\mathscr{A}_k is zero and we show that this is the case if and only if Ak\mathscr{A}_k is empty

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