Integers for which Sigma of Phi equals Sigma

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Theorem

The following positive integers have the property that the $\sigma$ value of their Euler $\phi$ value equals their $\sigma$ value:

$\sigma \left({\phi \left({n}\right)}\right) = \sigma \left({n}\right)$
$1, 87, 362, 1257, 1798, 5002, 9374, \ldots$

This sequence is A033631 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle \sigma \left({\phi \left({1}\right)}\right)\) \(=\) \(\displaystyle \sigma \left({1}\right)\) $\phi$ of $1$
\(\displaystyle \) \(=\) \(\displaystyle 1\) $\sigma$ of $1$


\(\displaystyle \sigma \left({\phi \left({87}\right)}\right)\) \(=\) \(\displaystyle \sigma \left({56}\right)\) $\phi$ of $87$
\(\displaystyle \) \(=\) \(\displaystyle 120\) $\sigma$ of $56$
\(\displaystyle \) \(=\) \(\displaystyle \sigma \left({87}\right)\) $\sigma$ of $87$


\(\displaystyle \sigma \left({\phi \left({362}\right)}\right)\) \(=\) \(\displaystyle \sigma \left({180}\right)\) $\phi$ of $362$
\(\displaystyle \) \(=\) \(\displaystyle 546\) $\sigma$ of $180$
\(\displaystyle \) \(=\) \(\displaystyle \sigma \left({362}\right)\) $\sigma$ of $362$


\(\displaystyle \sigma \left({\phi \left({1257}\right)}\right)\) \(=\) \(\displaystyle \sigma \left({836}\right)\) $\phi$ of $1257$
\(\displaystyle \) \(=\) \(\displaystyle 1680\) $\sigma$ of $836$
\(\displaystyle \) \(=\) \(\displaystyle \sigma \left({1257}\right)\) $\sigma$ of $1257$


\(\displaystyle \sigma \left({\phi \left({1798}\right)}\right)\) \(=\) \(\displaystyle \sigma \left({840}\right)\) $\phi$ of $1798$
\(\displaystyle \) \(=\) \(\displaystyle 2880\) $\sigma$ of $840$
\(\displaystyle \) \(=\) \(\displaystyle \sigma \left({1798}\right)\) $\sigma$ of $1798$


\(\displaystyle \sigma \left({\phi \left({5002}\right)}\right)\) \(=\) \(\displaystyle \sigma \left({2400}\right)\) $\phi$ of $5002$
\(\displaystyle \) \(=\) \(\displaystyle 7812\) $\sigma$ of $2400$
\(\displaystyle \) \(=\) \(\displaystyle \sigma \left({5002}\right)\) $\sigma$ of $5002$


\(\displaystyle \sigma \left({\phi \left({9374}\right)}\right)\) \(=\) \(\displaystyle \sigma \left({4536}\right)\) $\phi$ of $9374$
\(\displaystyle \) \(=\) \(\displaystyle 14 \, 520\) $\sigma$ of $4536$
\(\displaystyle \) \(=\) \(\displaystyle \sigma \left({9374}\right)\) $\sigma$ of $9374$


Sources