Question 71
Let $k\gt1$ be an integer. A function $f:\mathbb N^*\to\mathbb N^*$ is called $k$-tastrophic if $f_k(n)=n^k$ for every integer $n\gt0$, where $f_k$ is the $k$th iterate of $f$:
$$f_k(n)=\underbrace{f\circ f\circ\cdots\circ f}_{k\ \text{times}}(n).$$
For which $k$ does a $k$-tastrophic function exist?
$$f_k(n)=\underbrace{f\circ f\circ\cdots\circ f}_{k\ \text{times}}(n).$$
For which $k$ does a $k$-tastrophic function exist?