# 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 $\map A n$ denote the arithmetic mean of the divisors of $n$.

Then we have:

$\map A n = \dfrac {\map {\sigma_1} n} {\map {\sigma_0} n}$

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 $n$.

 $\ds \map {\sigma_1} 6$ $=$ $\ds 12$ $\sigma_1$ of $6$ $\ds \map {\sigma_0} 6$ $=$ $\ds 4$ $\sigma_0$ of $6$ $\ds \leadsto \ \$ $\ds \map A 6$ $=$ $\ds \dfrac {12} 4 = 3$ and so the arithmetic mean is an integer

 $\ds \map {\sigma_1} {28}$ $=$ $\ds 56$ $\sigma_1$ of $28$ $\ds \map {\sigma_0} {28}$ $=$ $\ds 4$ $\sigma_0$ of $28$ $\ds \leadsto \ \$ $\ds \map A {28}$ $=$ $\ds \dfrac {56} 6 = 9 \cdotp \dot 3$ and so the arithmetic mean is not an integer

 $\ds \map {\sigma_1} {140}$ $=$ $\ds 336$ $\sigma_1$ of $140$ $\ds \map {\sigma_0} {140}$ $=$ $\ds 12$ $\sigma_0$ of $140$ $\ds \leadsto \ \$ $\ds \map A {140}$ $=$ $\ds \dfrac {336} {12} = 28$ and so the arithmetic mean is an integer

 $\ds \map {\sigma_1} {270}$ $=$ $\ds 720$ $\sigma_1$ of $270$ $\ds \map {\sigma_0} {270}$ $=$ $\ds 16$ $\sigma_0$ of $270$ $\ds \leadsto \ \$ $\ds \map A {270}$ $=$ $\ds \dfrac {720} {16} = 45$ and so the arithmetic mean is an integer

 $\ds \map {\sigma_1} {496}$ $=$ $\ds 992$ $\sigma_1$ of $496$ $\ds \map {\sigma_0} {496}$ $=$ $\ds 10$ $\sigma_0$ of $496$ $\ds \leadsto \ \$ $\ds \map A {496}$ $=$ $\ds \dfrac {992} {10} = 9 \cdotp 92$ and so the arithmetic mean is not an integer

 $\ds \map {\sigma_1} {672}$ $=$ $\ds 2016$ $\sigma_1$ of $672$ $\ds \map {\sigma_0} {672}$ $=$ $\ds 24$ $\sigma_0$ of $672$ $\ds \leadsto \ \$ $\ds \map A {672}$ $=$ $\ds \dfrac {2016} {24} = 84$ and so the arithmetic mean is an integer

 $\ds \map {\sigma_1} {1638}$ $=$ $\ds 4368$ $\sigma_1$ of $1638$ $\ds \map {\sigma_0} {1638}$ $=$ $\ds 24$ $\sigma_0$ of $1638$ $\ds \leadsto \ \$ $\ds \map A {1638}$ $=$ $\ds \dfrac {4368} {24} = 182$ and so the arithmetic mean is an integer

$\blacksquare$