Definition:Ore Number
(Redirected from Definition:Harmonic Integer)
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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 $1948$.
Sources
- 1948: Øystein Ore: On the averages of the divisors of a number (Amer. Math. Monthly Vol. 55, no. 10: pp. 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. https://mathworld.wolfram.com/HarmonicDivisorNumber.html