Ore's conjecture

Statement

The conjecture states that the harmonic mean of all the positive divisors of an odd natural number greater than $1$ cannot be an integer.

The harmonic mean of all the positive divisors is given by the expression:

$\frac{n{\sigma }_0(n)}{{\sigma }_1(n)}$,

where ${\sigma }_0$ is the divisor count function and ${\sigma }_1$ is the divisor sum function. Natural numbers $n$ for which this ratio is an integer are termed harmonic divisor numbers or Ore numbers, and Ore's conjecture can thus be stated more compactly as: there is no odd Ore number.

Related facts and conjectures

Weaker conjectures

• Odd perfect number conjecture: The harmonic mean of the divisors of a perfect number is an integer, hence Ore's conjecture implies that there does not exist any odd perfect number.