PlanetMath: Ore number

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (22)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

Ore number

(Definition)

Given a positive integer with divisors $d_1, \ldots , d_k,$ if the harmonic mean

$\displaystyle {k \over {\sum_{i = 1}^k {1 \over {d_i}}}} \in \Bbb{Z},$

then

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 ${8 \over 3}$ . 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.

"Ore number" is owned by CompositeFan.

(view preamble | get metadata)

Other names:

harmonic divisor number

Log in to rate this entry.
(view current ratings)

Cross-references: odd, perfect numbers, even, OEIS, fraction, reciprocals, harmonic mean, divisors, integer, positive
There are 2 references to this entry.

This is version 12 of Ore number, born on 2006-06-02, modified 2006-09-15.
Object id is 7950, canonical name is OreNumber.
Accessed 1268 times total.

Classification:

AMS MSC:

11A05 (Number theory :: Elementary number theory :: Multiplicative structure; Euclidean algorithm; greatest common divisors)

Pending Errata and Addenda

None.[ View all 8 ]

Discussion

forum policy

Interact