On $\lambda(n)$.

Question

Let $\lambda(n)$ be the Carmichael Lambda Function of $n$.

We know that for every $m$ there is the largest $n_m\in\Bbb N$ such that for every $n>n_m$ we have $\lambda(n)\neq m$ and $\lambda(n_m)=n$ holds?

Given $m$ what is a good bound on $n_m$?