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 $\tau$ (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 $12$

$\tau \left({12}\right) = 6$


$\tau$ of $60$

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


$\tau$ of $105$

$\map \tau {105} = 8$


$\tau$ of $108$

$\tau \left({108}\right) = 12$


$\tau$ of $110$

$\tau \left({110}\right) = 8$


$\tau$ 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