# 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 $\sigma_0$ (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.