# Definition:Ore Number

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## Contents

## 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 $1948$.

## Sources

- 1948: Øystein Ore:
*On the averages of the divisors of a number*(*Amer. Math. Monthly***Vol. 55**,*no. 10*: 615 – 619) www.jstor.org/stable/2305616

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $140$

- Weisstein, Eric W. "Harmonic Divisor Number." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicDivisorNumber.html