Integers for which Divisor Sum of Phi equals Divisor Sum

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Theorem

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

$\map {\sigma_1} {\map \phi n} = \map {\sigma_1} n$
$1, 87, 362, 1257, 1798, 5002, 9374, \ldots$

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


Proof

\(\ds \map {\sigma_1} {\map \phi 1}\) \(=\) \(\ds \map {\sigma_1} 1\) $\phi$ of $1$
\(\ds \) \(=\) \(\ds 1\) $\sigma_1$ of $1$


\(\ds \map {\sigma_1} {\map \phi {87} }\) \(=\) \(\ds \map {\sigma_1} {56}\) $\phi$ of $87$
\(\ds \) \(=\) \(\ds 120\) $\sigma_1$ of $56$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {87}\) $\sigma_1$ of $87$


\(\ds \map {\sigma_1} {\map \phi {362} }\) \(=\) \(\ds \map {\sigma_1} {180}\) $\phi$ of $362$
\(\ds \) \(=\) \(\ds 546\) $\sigma_1$ of $180$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {362}\) $\sigma_1$ of $362$


\(\ds \map {\sigma_1} {\map \phi {1257} }\) \(=\) \(\ds \map {\sigma_1} {836}\) $\phi$ of $1257$
\(\ds \) \(=\) \(\ds 1680\) $\sigma_1$ of $836$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {1257}\) $\sigma_1$ of $1257$


\(\ds \map {\sigma_1} {\map \phi {1798} }\) \(=\) \(\ds \map {\sigma_1} {840}\) $\phi$ of $1798$
\(\ds \) \(=\) \(\ds 2880\) $\sigma_1$ of $840$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {1798}\) $\sigma_1$ of $1798$


\(\ds \map {\sigma_1} {\map \phi {5002} }\) \(=\) \(\ds \map {\sigma_1} {2400}\) $\phi$ of $5002$
\(\ds \) \(=\) \(\ds 7812\) $\sigma_1$ of $2400$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {5002}\) $\sigma_1$ of $5002$


\(\ds \map {\sigma_1} {\map \phi {9374} }\) \(=\) \(\ds \map {\sigma_1} {4536}\) $\phi$ of $9374$
\(\ds \) \(=\) \(\ds 14 \, 520\) $\sigma_1$ of $4536$
\(\ds \) \(=\) \(\ds \map {\sigma_1} {9374}\) $\sigma_1$ of $9374$


Sources

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