# 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

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $42$