# Definition:Ore Number

## Definition

Let $n \in \Z_{>0}$ be a positive integer.

$n$ is an Ore number if and only if the harmonic mean of its divisors is an integer.

## Sequence of Ore Numbers

The sequence of Ore numbers begins:

$1, 6, 28, 140, 270, 496, 672, 1638, 2970, 6200, 8128, 8190, 18600, 18620, \ldots$

## Examples

Let $H \left({n}\right)$ denote the harmonic mean of the divisors of $n$.

### $6$ is an Ore Number

$H \left({6}\right) = 2$

### $28$ is an Ore Number

$H \left({28}\right) = 3$

### $140$ is an Ore Number

$H \left({140}\right) = 5$

### $270$ is an Ore Number

$H \left({270}\right) = 6$

### $496$ is an Ore Number

$H \left({496}\right) = 5$

### $672$ is an Ore Number

$H \left({672}\right) = 8$

### $1638$ is an Ore Number

$H \left({1638}\right) = 9$

## Also known as

An Ore number can also be seen referred to as a harmonic number, but that term is used on $\mathsf{Pr} \infty \mathsf{fWiki}$ to denote a summation of reciprocals.

Other sources use the more unwieldy term harmonic divisor number.

Still others use the term Ore harmonic number or Ore's harmonic number.

## Source of Name

This entry was named for Øystein Ore.

## Historical Note

Ore numbers were defined by Øystein Ore in $1948$.