2014 OIM Problems/Problem 6

Problem

Given a set $X$ and a function $f: X \to X$ , we say that for each $x \in X$ , $f^1(x)=f(x)$ , and for each $j \ge 1$ , $f^{j+1}=f(f^j(x))$ . We say that $a \in X$ is a fixed point of $f$ if $f(a) = a$ . For each real number $x$ , we define $\pi (x)$ as the number of smaller positive primes less or equal to $x$ . Given a positive integer $n$ , we say that $f : \left\{1, 2, \cdots , n\right\} \to \left\{1, 2, c\dots , n\right\}$ it's "catracha" if $f^{f(k)}(k)=k$ for all $k \in \left\{ 1,2,\cdots,n\right\}$ Prove: