Perfect Number is Ore Number

Theorem
Let $n \in \Z_{>0}$ be a perfect number.

Then $n$ is an Ore number.

Proof
From Harmonic Mean of Divisors in terms of Divisor Count and Divisor Sum, the harmonic mean of the divisors of $n$ is given by:
 * $\map H n = \dfrac {n \map {\sigma_0} n} {\map {\sigma_1} n}$

where:
 * $\map {\sigma_0} n$ denotes the divisor counting function: the number of divisors of $n$
 * $\map {\sigma_1} n$ denotes the divisor sum function: the sum of the divisors of $n$.

Let $n$ be a perfect number.

By definition of perfect number:
 * $\dfrac {\map {\sigma_1} n} n = 2$

From Divisor Counting Function is Odd Iff Argument is Square:
 * $\map {\sigma_0} n = 2 k$

for some $k \in \Z$.

Hence:
 * $\map H n = \dfrac {2 k} 2 = k$

Hence the result.