# $n$-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 $(a_{1},\dots,a_{n-1})\in S^{n-1}$ with $\gcd(a_{i},a_{j})=1$ for $i\neq j$ such that $\displaystyle m=\prod_{j=1}^{n-1}{a_{j}}^{j}$.

Title $n$-free number NfreeNumber 2013-03-22 16:02:22 2013-03-22 16:02:22 Wkbj79 (1863) Wkbj79 (1863) 6 Wkbj79 (1863) Definition msc 11A51 SquareFreeNumber NFullNumber cubefree cubefree number cube free cube free number cube-free cube-free number