$n$ -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
Canonical name	NfreeNumber
Date of creation	2013-03-22 16:02:22
Last modified on	2013-03-22 16:02:22
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	6
Author	Wkbj79 (1863)
Entry type	Definition
Classification	msc 11A51
Related topic	SquareFreeNumber
Related topic	NFullNumber
Defines	cubefree
Defines	cubefree number
Defines	cube free
Defines	cube free number
Defines	cube-free
Defines	cube-free number

n<math id="m1" class="ltx_Math" alttext="n" display="inline"><mi>n</mi></math>-free number

$n$ -free number