# Numbers such that Tau divides Phi divides Sigma

## Theorem

The sequence of integers $n$ with the property that:

$\map \tau n \divides \map \phi n \divides \map \sigma n$

where:

$\divides$ denotes divisibility
$\tau$ denotes the $\tau$ (tau) function: the count of divisors of $n$
$\phi$ denotes the Euler $\phi$ (phi) function: the count of smaller integers coprime to $n$
$\sigma$ denotes the $\sigma$ (sigma) function: the sum of divisors of $n$

begins:

$1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, \ldots$

## Proof

By inspection and investigation.

$\blacksquare$

## Examples

### 1

 $\displaystyle \map \tau 1$ $=$ $\, \displaystyle 1 \,$ $\displaystyle$ $\tau$ of $1$ $\displaystyle \map \phi 1$ $=$ $\, \displaystyle 1 \,$ $\displaystyle$ $\phi$ of $1$ $\displaystyle \map \sigma 1$ $=$ $\, \displaystyle 1 \,$ $\displaystyle$ $\sigma$ of $1$

$\blacksquare$

### 3

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

$\blacksquare$

### 15

 $\displaystyle \tau \left({15}\right)$ $=$ $\, \displaystyle 4 \,$ $\displaystyle$ $\tau$ of $15$ $\displaystyle \phi \left({15}\right)$ $=$ $\, \displaystyle 8 \,$ $\, \displaystyle =\,$ $\displaystyle 2 \times 4$ $\phi$ of $15$ $\displaystyle \sigma \left({15}\right)$ $=$ $\, \displaystyle 24 \,$ $\, \displaystyle =\,$ $\displaystyle 3 \times 8$ $\sigma$ of $15$

$\blacksquare$

### 30

 $\displaystyle \tau \left({30}\right)$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $30$ $\displaystyle \phi \left({30}\right)$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\phi$ of $30$ $\displaystyle \sigma \left({30}\right)$ $=$ $\, \displaystyle 72 \,$ $\, \displaystyle =\,$ $\displaystyle 9 \times 8$ $\sigma$ of $30$

$\blacksquare$

### 35

 $\displaystyle \tau \left({35}\right)$ $=$ $\, \displaystyle 4 \,$ $\displaystyle$ $\tau$ of $35$ $\displaystyle \phi \left({35}\right)$ $=$ $\, \displaystyle 24 \,$ $\, \displaystyle =\,$ $\displaystyle 6 \times 4$ $\phi$ of $35$ $\displaystyle \sigma \left({35}\right)$ $=$ $\, \displaystyle 48 \,$ $\, \displaystyle =\,$ $\displaystyle 2 \times 24$ $\sigma$ of $35$

$\blacksquare$

### 56

 $\displaystyle \tau \left({56}\right)$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $56$ $\displaystyle \phi \left({56}\right)$ $=$ $\, \displaystyle 24 \,$ $\, \displaystyle =\,$ $\displaystyle 3 \times 8$ $\phi$ of $56$ $\displaystyle \sigma \left({56}\right)$ $=$ $\, \displaystyle 120 \,$ $\, \displaystyle =\,$ $\displaystyle 5 \times 24$ $\sigma$ of $56$

$\blacksquare$

### 70

 $\displaystyle \map \tau {70}$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $70$ $\displaystyle \map \phi {70}$ $=$ $\, \displaystyle 24 \,$ $\, \displaystyle =\,$ $\displaystyle 3 \times 8$ $\phi$ of $70$ $\displaystyle \map \sigma {70}$ $=$ $\, \displaystyle 144 \,$ $\, \displaystyle =\,$ $\displaystyle 6 \times 24$ $\sigma$ of $70$

$\blacksquare$

### 78

 $\displaystyle \map \tau {78}$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $78$ $\displaystyle \map \phi {78}$ $=$ $\, \displaystyle 24 \,$ $\, \displaystyle =\,$ $\displaystyle 3 \times 8$ $\phi$ of $78$ $\displaystyle \map \sigma {78}$ $=$ $\, \displaystyle 168 \,$ $\, \displaystyle =\,$ $\displaystyle 7 \times 24$ $\sigma$ of $78$

$\blacksquare$

### 105

 $\displaystyle \map \tau {105}$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $105$ $\displaystyle \map \phi {105}$ $=$ $\, \displaystyle 48 \,$ $\, \displaystyle =\,$ $\displaystyle 6 \times 8$ $\phi$ of $105$ $\displaystyle \map \sigma {105}$ $=$ $\, \displaystyle 192 \,$ $\, \displaystyle =\,$ $\displaystyle 4 \times 48$ $\sigma$ of $105$

$\blacksquare$

### 140

 $\displaystyle \tau \left({140}\right)$ $=$ $\, \displaystyle 12 \,$ $\displaystyle$ $\tau$ of $140$ $\displaystyle \phi \left({140}\right)$ $=$ $\, \displaystyle 48 \,$ $\, \displaystyle =\,$ $\displaystyle 4 \times 12$ $\phi$ of $140$ $\displaystyle \sigma \left({140}\right)$ $=$ $\, \displaystyle 336 \,$ $\, \displaystyle =\,$ $\displaystyle 7 \times 48$ $\sigma$ of $140$

$\blacksquare$

### 168

 $\displaystyle \tau \left({168}\right)$ $=$ $\, \displaystyle 16 \,$ $\displaystyle$ $\tau$ of $168$ $\displaystyle \phi \left({168}\right)$ $=$ $\, \displaystyle 48 \,$ $\, \displaystyle =\,$ $\displaystyle 3 \times 16$ $\phi$ of $168$ $\displaystyle \sigma \left({168}\right)$ $=$ $\, \displaystyle 480 \,$ $\, \displaystyle =\,$ $\displaystyle 10 \times 48$ $\sigma$ of $168$

$\blacksquare$

### 190

 $\displaystyle \tau \left({190}\right)$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $190$ $\displaystyle \phi \left({190}\right)$ $=$ $\, \displaystyle 72 \,$ $\, \displaystyle =\,$ $\displaystyle 9 \times 8$ $\phi$ of $190$ $\displaystyle \sigma \left({190}\right)$ $=$ $\, \displaystyle 360 \,$ $\, \displaystyle =\,$ $\displaystyle 5 \times 72$ $\sigma$ of $190$

$\blacksquare$

### 210

 $\displaystyle \tau \left({210}\right)$ $=$ $\, \displaystyle 16 \,$ $\displaystyle$ $\tau$ of $210$ $\displaystyle \phi \left({210}\right)$ $=$ $\, \displaystyle 48 \,$ $\, \displaystyle =\,$ $\displaystyle 3 \times 16$ $\phi$ of $210$ $\displaystyle \sigma \left({210}\right)$ $=$ $\, \displaystyle 576 \,$ $\, \displaystyle =\,$ $\displaystyle 12 \times 48$ $\sigma$ of $210$

$\blacksquare$

### 248

 $\displaystyle \tau \left({248}\right)$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $248$ $\displaystyle \phi \left({248}\right)$ $=$ $\, \displaystyle 120 \,$ $\, \displaystyle =\,$ $\displaystyle 15 \times 8$ $\phi$ of $248$ $\displaystyle \sigma \left({248}\right)$ $=$ $\, \displaystyle 480 \,$ $\, \displaystyle =\,$ $\displaystyle 4 \times 120$ $\sigma$ of $248$

$\blacksquare$

### 264

 $\displaystyle \tau \left({264}\right)$ $=$ $\, \displaystyle 16 \,$ $\displaystyle$ $\tau$ of $264$ $\displaystyle \phi \left({264}\right)$ $=$ $\, \displaystyle 80 \,$ $\, \displaystyle =\,$ $\displaystyle 5 \times 16$ $\phi$ of $264$ $\displaystyle \sigma \left({264}\right)$ $=$ $\, \displaystyle 720 \,$ $\, \displaystyle =\,$ $\displaystyle 9 \times 80$ $\sigma$ of $264$

$\blacksquare$

### 357

 $\displaystyle \tau \left({357}\right)$ $=$ $\, \displaystyle 8 \,$ $\displaystyle$ $\tau$ of $357$ $\displaystyle \phi \left({357}\right)$ $=$ $\, \displaystyle 192 \,$ $\, \displaystyle =\,$ $\displaystyle 24 \times 8$ $\phi$ of $357$ $\displaystyle \sigma \left({357}\right)$ $=$ $\, \displaystyle 576 \,$ $\, \displaystyle =\,$ $\displaystyle 3 \times 192$ $\sigma$ of $357$

$\blacksquare$

### 420

 $\displaystyle \tau \left({420}\right)$ $=$ $\, \displaystyle 24 \,$ $\displaystyle$ $\tau$ of $420$ $\displaystyle \phi \left({420}\right)$ $=$ $\, \displaystyle 96 \,$ $\, \displaystyle =\,$ $\displaystyle 4 \times 24$ $\phi$ of $420$ $\displaystyle \sigma \left({420}\right)$ $=$ $\, \displaystyle 1344 \,$ $\, \displaystyle =\,$ $\displaystyle 14 \times 96$ $\sigma$ of $420$

$\blacksquare$

### 570

 $\displaystyle \tau \left({570}\right)$ $=$ $\, \displaystyle 16 \,$ $\displaystyle$ $\tau$ of $570$ $\displaystyle \phi \left({570}\right)$ $=$ $\, \displaystyle 144 \,$ $\, \displaystyle =\,$ $\displaystyle 9 \times 16$ $\phi$ of $570$ $\displaystyle \sigma \left({570}\right)$ $=$ $\, \displaystyle 1440 \,$ $\, \displaystyle =\,$ $\displaystyle 10 \times 144$ $\sigma$ of $570$

$\blacksquare$

### 616

 $\displaystyle \tau \left({616}\right)$ $=$ $\, \displaystyle 16 \,$ $\displaystyle$ $\tau$ of $616$ $\displaystyle \phi \left({616}\right)$ $=$ $\, \displaystyle 240 \,$ $\, \displaystyle =\,$ $\displaystyle 15 \times 16$ $\phi$ of $616$ $\displaystyle \sigma \left({616}\right)$ $=$ $\, \displaystyle 1440 \,$ $\, \displaystyle =\,$ $\displaystyle 6 \times 240$ $\sigma$ of $616$

$\blacksquare$

### 630

 $\displaystyle \map \tau {630}$ $=$ $\, \displaystyle 24 \,$ $\displaystyle$ $\tau$ of $630$ $\displaystyle \map \phi {630}$ $=$ $\, \displaystyle 144 \,$ $\, \displaystyle =\,$ $\displaystyle 6 \times 24$ $\phi$ of $630$ $\displaystyle \map \sigma {630}$ $=$ $\, \displaystyle 1872 \,$ $\, \displaystyle =\,$ $\displaystyle 13 \times 144$ $\sigma$ of $630$

$\blacksquare$