The iterated Carmichael lambda function

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 $k.$