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

Abstract

For each positive integer $k$, let ${\mathcal{A}}_k$ be the set of all positive integers $n$ such that $\gcd (n,F_n)=k$, where $F_n$ denotes the $n$th Fibonacci number. We prove that the asymptotic density of ${\mathcal{A}}_k$ exists and is equal to $$\sum _{d=1}^{\infty }\frac{\mu (d)}{\mathrm{lcm}(dk,z(dk))}$$ where $\mu$ is the Möbius function and $z(m)$ denotes the least positive integer $n$ such that $m$ divides $F_n$. We also give an effective criterion to establish when the asymptotic density of ${\mathcal{A}}_k$ is zero and we show that this is the case if and only if ${\mathcal{A}}_k$ is empty.

References in corpus (2)

• On the greatest common divisor of $n$ and the $n$th Fibonacci number
• On numbers $n$ relatively prime to the $n$th term of a linear recurrence

Cited by in corpus (2)

• On numbers $n$ with polynomial image coprime with the $n$th term of a linear recurrence
• Greatest common divisors of shifted primes and Fibonacci numbers