# Definition:Divisor Counting Function

## Definition

Let $n$ be an integer such that $n \ge 1$.

The **divisor counting function** is defined on $n$ as being the total number of positive integer divisors of $n$.

It is denoted on $\mathsf{Pr} \infty \mathsf{fWiki}$ as $\tau$ (the Greek letter **tau**).

That is:

- $\displaystyle \map \tau n = \sum_{d \mathop \divides n} 1$

where $\displaystyle \sum_{d \mathop \divides n}$ is the sum over all divisors of $n$.

## Also known as

Some sources refer to this as the **divisor function** and denote it $\map d n$, but as there is a more general definition of the divisor function the more precise name **divisor counting function** is preferred.

It is also often referred to as the **$\tau$ (tau) function**, but there are a number of functions with such a name.

Hence the unwieldy, but practical, **divisor counting function**, which is non-standard.

## Examples

### $\tau$ of $1$

The value of the $\tau$ function for the integer $1$ is $1$.

### $\tau$ of $3$

- $\map \tau 3 = 2$

### $\tau$ of $12$

- $\map \tau {12} = 6$

### $\tau$ of $60$

- $\tau \left({60}\right) = 12$

### $\tau$ of $105$

- $\map \tau {105} = 8$

### $\tau$ of $108$

- $\map \tau {108} = 12$

### $\tau$ of $110$

- $\map \tau {110} = 8$

### $\tau$ of $120$

- $\tau \left({120}\right) = 16$

## Also see

- Results about
**the $\tau$ function**can be found here.

## Linguistic Note

The name of the Greek letter $\tau$ (tau) is properly pronounced to rhyme with **cow** not **caw**.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**divisor function**