# $n$-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 $\mathrm{gcd}({a}_{i},{a}_{j})=1$ for $i\ne j$ such that $m={\displaystyle \prod _{j=1}^{n-1}}a_{j}{}^{j}$.

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 |