Integers whose Ratio between Sigma and Phi is Square

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Theorem

The sequence of integers whose $\sigma$ value divided by its Euler $\phi$ value is a square begins:

$1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, \ldots$

This sequence is A293391 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).



Proof


\(\displaystyle \map \phi {714}\) \(=\) \(\displaystyle 192\) $\phi$ of $714$
\(\displaystyle \map \sigma {714}\) \(=\) \(\displaystyle 1728\) $\sigma$ of $714$
\(\displaystyle \map \sigma {714} / \map \phi {714}\) \(=\) \(\displaystyle 1728 / 192\)
\(\displaystyle \) \(=\) \(\displaystyle 9\)
\(\displaystyle \) \(=\) \(\displaystyle 3^2\)



Sources