# Definition:Divisor Counting Function

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## 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 $\sigma_0$ (the Greek letter **sigma**).

That is:

- $\ds \map {\sigma_0} n = \sum_{d \mathop \divides n} 1$

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

## Also known as

Some sources refer to this as the **divisor function** and denote it $\map d n$.

However, as this function is an instance of 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.

Some sources use $\nu$, but again, that also has multiple uses.

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

## Examples

### $\sigma_0$ of $1$

The value of the divisor counting function for the integer $1$ is $1$.

### $\sigma_0$ of $3$

- $\map {\sigma_0} 3 = 2$

### $\sigma_0$ of $12$

- $\map {\sigma_0} {12} = 6$

### $\sigma_0$ of $60$

- $\map {\sigma_0} {60} = 12$

### $\sigma_0$ of $105$

- $\map {\sigma_0} {105} = 8$

### $\sigma_0$ of $108$

- $\map {\sigma_0} {108} = 12$

### $\sigma_0$ of $110$

- $\map {\sigma_0} {110} = 8$

### $\sigma_0$ of $120$

- $\map {\sigma_0} {120} = 16$

## Also see

- Results about
**the divisor count function**can be found**here**.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**divisor function** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**divisor function** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**divisor function**