# Integers whose Phi times Tau equal Sigma

## Theorem

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

$1, 3, 14, 42$

## 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 $42$