Ore's conjecture

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This conjecture states that there are no odd numbers satisfying a certain condition. Typically, there are known to be many even numbers (in some cases, there are known to be infinitely many even numbers) satisfying the condition.
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Statement

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

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

,

where is the divisor count function and is the divisor sum function. Natural numbers 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

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