Some divisibilities amongst the terms of linear recurrences

Let (un)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) orΩ(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 integersn 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.14 page(s