Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors

Theorem
The following sequence of integers have the property that both the harmonic mean and arithmetic mean of their divisors are integers:
 * $1, 6, 140, 270, 672, \ldots$

Proof
Note the integers whose harmonic mean of their divisors are integers are the Ore numbers:
 * $1, 6, 28, 140, 270, 496, 672, \ldots$

It remains to calculate the arithmetic mean of their divisors.

Let $A \left({n}\right)$ denote the arithmetic mean of the divisors of $n$.

Then we have:
 * $A \left({n}\right) = \dfrac {\sigma \left({n}\right)} {\tau \left({n}\right)}$

where:
 * $\tau \left({n}\right)$ denotes the $\tau$ (tau) function: the number of divisors of $n$
 * $\sigma \left({n}\right)$ denotes the $\sigma$ (sigma) function: the sum of the divisors of $n$.