For each positive integer k, let Ak be the set of all positive integers n such that gcd(n, Fn) = k, whereFn denotes the nth Fibonacci number. We prove that the asymptotic density of Ak exists and is equal to ∑∞d=1µ(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 thisis the case if and only if Akis empty

The density of numbers n having a prescribed G.C.D. with the nth Fibonacci number / Sanna, Carlo; Tron, Emanuele. - In: INDAGATIONES MATHEMATICAE. - ISSN 0019-3577. - STAMPA. - 29:3(2018), pp. 972-980. [10.1016/j.indag.2018.03.002]

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

Sanna, Carlo;
2018

Abstract

For each positive integer k, let Ak be the set of all positive integers n such that gcd(n, Fn) = k, whereFn denotes the nth Fibonacci number. We prove that the asymptotic density of Ak exists and is equal to ∑∞d=1µ(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 thisis the case if and only if Akis empty
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11583/2722600