# 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 $n$.

 $\displaystyle \sigma \left({6}\right)$ $=$ $\displaystyle 12$ $\sigma$ of $6$ $\displaystyle \tau \left({6}\right)$ $=$ $\displaystyle 4$ $\tau$ of $6$ $\displaystyle \leadsto \ \$ $\displaystyle A \left({6}\right)$ $=$ $\displaystyle \dfrac {12} 4 = 3$ and so the arithmetic mean is an integer

 $\displaystyle \sigma \left({28}\right)$ $=$ $\displaystyle 56$ $\sigma$ of $28$ $\displaystyle \tau \left({28}\right)$ $=$ $\displaystyle 4$ $\tau$ of $28$ $\displaystyle \leadsto \ \$ $\displaystyle A \left({28}\right)$ $=$ $\displaystyle \dfrac {56} 6 = 9 \cdotp \dot 3$ and so the arithmetic mean is not an integer

 $\displaystyle \sigma \left({140}\right)$ $=$ $\displaystyle 336$ $\sigma$ of $140$ $\displaystyle \tau \left({140}\right)$ $=$ $\displaystyle 12$ $\tau$ of $140$ $\displaystyle \leadsto \ \$ $\displaystyle A \left({140}\right)$ $=$ $\displaystyle \dfrac {336} {12} = 28$ and so the arithmetic mean is an integer

 $\displaystyle \sigma \left({270}\right)$ $=$ $\displaystyle 720$ $\sigma$ of $270$ $\displaystyle \tau \left({270}\right)$ $=$ $\displaystyle 16$ $\tau$ of $270$ $\displaystyle \leadsto \ \$ $\displaystyle A \left({270}\right)$ $=$ $\displaystyle \dfrac {720} {16} = 45$ and so the arithmetic mean is an integer

 $\displaystyle \sigma \left({496}\right)$ $=$ $\displaystyle 992$ $\sigma$ of $496$ $\displaystyle \tau \left({496}\right)$ $=$ $\displaystyle 10$ $\tau$ of $496$ $\displaystyle \leadsto \ \$ $\displaystyle A \left({496}\right)$ $=$ $\displaystyle \dfrac {992} {10} = 9 \cdotp 92$ and so the arithmetic mean is not an integer

 $\displaystyle \sigma \left({672}\right)$ $=$ $\displaystyle 2016$ $\sigma$ of $672$ $\displaystyle \tau \left({672}\right)$ $=$ $\displaystyle 24$ $\tau$ of $672$ $\displaystyle \leadsto \ \$ $\displaystyle A \left({672}\right)$ $=$ $\displaystyle \dfrac {2016} {24} = 84$ and so the arithmetic mean is an integer

 $\displaystyle \sigma \left({1638}\right)$ $=$ $\displaystyle 4368$ $\sigma$ of $1638$ $\displaystyle \tau \left({1638}\right)$ $=$ $\displaystyle 24$ $\tau$ of $1638$ $\displaystyle \leadsto \ \$ $\displaystyle A \left({1638}\right)$ $=$ $\displaystyle \dfrac {4368} {24} = 182$ and so the arithmetic mean is an integer

$\blacksquare$