# Ore number

Given a positive integer $n$ with divisors^{} ${d}_{1},\mathrm{\dots},{d}_{k},$ if the harmonic mean

$$\frac{k}{{\sum}_{i=1}^{k}\frac{1}{{d}_{i}}}\in \mathbb{Z},$$ |

then $n$ is an Ore number or harmonic divisor number.

For example, 270 has the divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135 and 270. The reciprocals of these 16 divisors add up to $\frac{8}{3}$. Then 16 divided by that fraction is 6, an integer. Thus 270 is an Ore number.

The first few Ore numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, listed in A001599 of Sloane’s OEIS.

All even perfect numbers are Ore numbers, a fact proven by Øystein Ore in 1948.

1 is the only known odd Ore number. If there’s another, it would have to be pretty big, and is considered as unlikely to exist as an odd perfect number.

Title | Ore number |
---|---|

Canonical name | OreNumber |

Date of creation | 2013-03-22 15:56:28 |

Last modified on | 2013-03-22 15:56:28 |

Owner | CompositeFan (12809) |

Last modified by | CompositeFan (12809) |

Numerical id | 15 |

Author | CompositeFan (12809) |

Entry type | Definition |

Classification | msc 11A05 |

Synonym | harmonic divisor number |