Some divisibilities amongst the terms of linear recurrences (Q998238)

Consider an integer linear recurrence sequence \((u_n)_{n \geq 0}\) of order \(r \geq 2\): \[ u_{n + r} = a_1u_{n + r - 1} + \dots + a_{r - 1}u_{n + 1} + a_ru_n \quad (n \geq 0), \] and let \(\psi(x)\) be its characteristic polynomial. The sequence \((u_n)_{n \geq 0}\) is called \textit{non-degenerate}, if \(\lambda^m \neq \mu^m\) for any \(m \in \mathbb N\) and any two distinct roots \(\lambda, \mu\) of \(\psi(x)\). Given an arithmetic function \(f\), let \(N_f(x)\) denote the number of integers \(n\), with \(n \leq x\), such that \(f(n) \mid u_n\). In this paper, the authors obtain bounds of the form \(N_f(x) = o(x)\) when \(f(n)\) is one of the functions \(\tau(n), \omega(n)\), or \(\Omega(n)\) (these are, respectively, the number of positive divisors of \(n\), the number of distinct prime divisors of \(n\), and the total number of prime divisors of \(n\)). Suppose that the sequence \((u_n)_{n \geq 0}\) is non-degenerate. When \(f(n) = \omega(n)\) or \(f(n) = \Omega(n)\), the authors prove that \(N_f(x) \ll x(\log\log\log x)^{-c_r}\), where \(c_r > 0\) is a constant described explicitly in the paper. They also show that when \(a_r\) is odd, one has \(N_{\tau}(x) \ll x(\log x)^{-2^{-r}}\). The authors note that the condition on \(a_r\) in the latter result cannot be entirely dispensed with.