This is just a curiosity; as far as I know it is of no deep mathematical significance.
Consider the function $f : \mathbb Z_{> 0} \to \mathbb Z_{> 0}$ defined by putting $f(1) = 1$, and $$ f\Bigl(\prod_{i = 1}^n p_i^{a_i}\Bigr) = \prod_{i = 1}^n a_i^{p_i} $$ whenever $n$ is a positive integer, $p_1, \dotsc, p_n$ are distinct primes, and $a_1, \dotsc, a_n$ are positive integers. For example, $f(n) = 1$ whenever $n$ is square-free; $f(4) = 4$; $f(8) = 9$ and $f(9) = 8$; $f(16) = 16$; and $f(25) = 32$ and $f(32) = 25$.
What, if anything, is known about $f$? For example, does it have any unbounded orbits? What is its image? What are its periods (i.e., the positive integers $p$ so that there is some integer $n$ for which $f^p(n) = n$ but $f^q(n) \ne n$ for all positive integers $q$ with $q < p$)?