Integers whose Phi times Tau equal Sigma

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Theorem

The positive integers whose Euler $\phi$ function multiplied by its $\tau$ function equals its $\sigma$ function are:

$1, 3, 14, 42$

This sequence is A104905 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Proof

\(\displaystyle \map \phi 1 \map \tau 1\) \(=\) \(\displaystyle 1 \times 1\) $\phi$ of $1$, $\tau$ of $1$
\(\displaystyle \) \(=\) \(\displaystyle 1\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sigma 1\) $\sigma$ of $1$


\(\displaystyle \map \phi 3 \map \tau 3\) \(=\) \(\displaystyle 2 \times 2\) $\phi$ of $3$, $\tau$ of $3$
\(\displaystyle \) \(=\) \(\displaystyle 4\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sigma 3\) $\sigma$ of $3$


\(\displaystyle \map \phi {14} \map \tau {14}\) \(=\) \(\displaystyle 6 \times 4\) $\phi$ of $14$, $\tau$ of $14$
\(\displaystyle \) \(=\) \(\displaystyle 24\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sigma {14}\) $\sigma$ of $14$


\(\displaystyle \map \phi {42} \map \tau {42}\) \(=\) \(\displaystyle 12 \times 8\) $\phi$ of $42$, $\tau$ of $42$
\(\displaystyle \) \(=\) \(\displaystyle 96\)
\(\displaystyle \) \(=\) \(\displaystyle \map \sigma {42}\) $\sigma$ of $42$



Sources