On numbers n dividing the nth term of a Lucas sequence

We prove that if $(u_n)_{n \geq 0}$ is a Lucas sequence satisfying some mild hypotheses, then the number of positive integers $n$ not exceeding $x$ and such that $n$ divides $u_n$ is less than \begin{equation*} x^{1-(1/2+o(1)) \log \log \log x / \log \log x} , \end{equation*} as $x \to \infty$. This both generalizes a result of Luca and Tron about the positive integers $n$ dividing the $n$-th Fibonacci number, and improves a previous upper bound due to Alba~Gonz\'{a}lez, Luca, Pomerance, and Shparlinski.