Some problems related to the growth of z(n)

Abstract

Let (Fn)n >= 0 be the Fibonacci sequence. The order of appearance z(n) of a positive integer n is defined as z(n):=min{k >= 1:n divide Fk}. In 2013, Marques proved that lim infn ->infinity z(n)/n=0. Let epsilon be a positive real number. In this paper, in particular, we generalized this Marques' result by proving that almost all positive integers satisfy z(n)/n < epsilon.