# 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:

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

## 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 $9374$