Ore number


Given a positive integer n with divisorsMathworldPlanetmathPlanetmath d1,,dk, if the harmonic mean

ki=1k1di,

then n 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 83. 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.

Title Ore number
Canonical name OreNumber
Date of creation 2013-03-22 15:56:28
Last modified on 2013-03-22 15:56:28
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 15
Author CompositeFan (12809)
Entry type Definition
Classification msc 11A05
Synonym harmonic divisor number