Definition:Ore Number

(Redirected from Definition:Harmonic Integer)

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 $\map H n$ denote the harmonic mean of the divisors of $n$.

$6$ is an Ore Number

$\map H 6 = 2$

$28$ is an Ore Number

$\map H {28} = 3$

$140$ is an Ore Number

$\map H {140} = 5$

$270$ is an Ore Number

$\map H {270} = 6$

$496$ is an Ore Number

$\map H {496} = 5$

$672$ is an Ore Number

$\map H {672} = 8$

$1638$ is an Ore Number

$\map H {1638} = 9$

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.

The term harmonic integer can also be found.

Source of Name

This entry was named for Øystein Ore.

Historical Note

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