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 $A \left({n}\right)$ 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

$\blacksquare$


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