Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors

From ProofWiki
Jump to navigation Jump to search

Theorem

The following sequence of integers have the property that both the harmonic mean and arithmetic mean of their divisors are integers:

$1, 6, 140, 270, 672, \ldots$

This sequence is A007340 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

Note the integers whose harmonic mean of their divisors are integers are the Ore numbers:

$1, 6, 28, 140, 270, 496, 672, \ldots$

This sequence is A001599 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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

$\blacksquare$


Sources