$n$-free number

The concept of a squarefree number can be generalized. Let $n\in \mathbb{Z}$ with $n>1$. Then $m\in \mathbb{Z}$ is $n$-free if, for any prime $p$, $p^n$ does not divide $m$.

Let $S$ denote the set of all squarefree natural numbers. Note that, for any $n$ and any positive $n$-free integer $m$, there exists a unique $(a_1,\mathrm{\dots },a_{n-1})\in S^{n-1}$ with $\gcd (a_i,a_j)=1$ for $i\ne j$ such that $m={\displaystyle \prod _{j=1}^{n-1}}a_j{}^j$.