The iterated Carmichael lambda function

Abstract

The Carmichael lambda function $\lambda (n)$ is defined to be the smallest positive integer $m$ such that $a^m$ is congruent to 1 modulo $n,$ for all $a$ and $n$ relatively prime. The function ${\lambda }_k(n)$ is defined to be the $k$th iterate of $\lambda (n).$ Previous results show a normal order for $n/{\lambda }_k(n)$ where $k=1,2.$ We will show a normal order for all $k.$