Explicit Prime Densities for the Rank of Appearance in Lucas Sequences

Abstract

Let $U$ be a Lucas sequence, $p$ be prime, and $ρ_U(p)$ be the rank of appearance of $p$ in $U$. We derive closed-form formulas for the Dirichlet density of primes $p$ for which $d\mid ρ_U(p)$, where $d\geq 1$ is a fixed integer. Our results complete the work of Sanna ($2022$) by covering all $U$ and all $d\geq 1$.