Ore's Conjecture

Define the Harmonic Mean of the Divisors of $n$

$$H(n)\equiv {\tau(n)\over \sum_{d\vert n} {1\over d}},$$

where $\tau(n)$ is the Tau Function (the number of Divisors of $n$). If $n$ is a Perfect Number, $H(n)$ is an Integer. Ore conjectured that if $n$ is Odd, then $H(n)$ is not an Integer. This implies that no Odd Perfect Numbers exist.

See also Harmonic Divisor Number, Harmonic Mean, Perfect Number, Tau Function