Some problems related to the growth of $z(n)$

Abstract

AbstractLet $(F_{n})_{n\geq 0}$(Fn)n≥0 be the Fibonacci sequence. The order of appearance $z(n)$z(n) of a positive integer n is defined as $z(n):= \min \{k\geq 1: n\mid F_{k}\}$z(n):=min{k≥1:n∣Fk}. In 2013, Marques proved that $\liminf_{n\to \infty }z(n)/n=0$lim infn→∞z(n)/n=0. Let ϵ 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 $z(n)/n<ϵ.