Explicit Prime Densities for Lucas Sequence Rank Divisibility

Let $U$ be a Lucas sequence, $p$ be a prime, and ${\rho }_U(p)$ be the rank of appearance of $p$ in $U$, that is, the least positive integer $n$ such that $p\mid U_n$, if it exists. We derive closed-form formulas for the Dirichlet density of primes $p$ for which $d\mid {\rho }_U(p)$, where $d\ge 1$ is a fixed integer. Our results complete the work of Sanna ($2022$) by covering all $U$ and all $d\ge 1$.