Editing Harmonic divisor number (section)

== Harmonic divisor numbers and perfect numbers ==
[[File:Perfect number Cuisenaire rods 6 exact.svg|thumb|Demonstration, with [[Cuisenaire rods]], of the perfection of the number 6]]

For any integer ''M'', as Ore observed, the product of the harmonic mean and [[arithmetic mean]] of its divisors equals ''M'' itself, as can be seen from the definitions. Therefore, ''M'' is harmonic, with harmonic mean of divisors ''k'', [[if and only if]] the average of its divisors is the product of ''M'' with a [[unit fraction]] 1/''k''.

Ore showed that every [[perfect number]] is harmonic. To see this, observe that the sum of the divisors of a perfect number ''M'' is exactly ''2M''; therefore, the average of the divisors is ''M''(2/τ(''M'')), where τ(''M'') denotes the [[Divisor function|number of divisors]] of ''M''. For any ''M'', τ(''M'') is [[parity (mathematics)|odd]] if and only if ''M'' is a [[square number]], for otherwise each divisor ''d'' of ''M'' can be paired with a different divisor ''M''/''d''. But no perfect number can be a square: this follows from the [[Perfect_number#Even_perfect_numbers|known form of even perfect numbers]] and from the fact that odd perfect numbers (if they exist) must have a factor of the form ''q''<sup>α</sup> where α ≡ 1 ([[modular arithmetic|mod]] 4). Therefore, for a perfect number ''M'', τ(''M'') is even and the average of the divisors is the product of ''M'' with the unit fraction 2/τ(''M''); thus, ''M'' is a harmonic divisor number.

Ore [[conjecture]]d that no odd harmonic divisor numbers exist other than 1. If the conjecture is true, this would imply the nonexistence of [[odd perfect number]]s.