Some divisibilities amongst the terms of linear recurrences

Let (u_n)_n≥0 be a non-degenerate linear recurrence sequence of integers. We show that the set of positive integers n such that either ω(n) or Ω(n) divides u_n is of asymptotic density zero, where ω(n) and Ω(n) are the numbers of prime and prime power divisors of n, respectively. The same also holds for the set of positive integers n such that τ(n) u_n, where τ(n) is the number of the positive integer divisors of n, provided that u_n satisfies some mild technical conditions.