Arithmetic properties of Apéry numbers

Abstract

Let (An)n⩾1 be the sequence of Apéry numbers with a general term given by A n = ∑ k n ( k n ) 2 ( k n + k ) 2 . In this paper, we prove that both the inequalities ω(An) > c0 log log log n and P(An) > c0 (log n log log n)1/2 hold for a set of positive integers n of asymptotic density 1. Here, ω(m) is the number of distinct prime factors of m, P(m) is the largest prime factor of m and c0 > 0 is an absolute constant. The method applies to more general sequences satisfying both a linear recurrence of order 2 with polynomial coefficients and certain Lucas-type congruences.