Resumen de Arithmetic properties of Apéry numbers

Florian Luca, Igor E. Shparlinski

• Let (An)n1 be the sequence of Apéry numbers with a general term given by . In this paper, we prove that both the inequalities (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, (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