Sequence of Numbers with Integer Arithmetic and Harmonic Means of Divisors
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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 count 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
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $140$