Definition:Tau Function

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Definition

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

The $\tau$ (tau) function is defined on $n$ as being the total number of positive integer divisors of $n$.

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 tau function is preferred.

It can also be referred to as the divisor counting function.


Examples

The $\tau$ function for the first few positive integers is as follows:

$\begin{array} {r|r} n & \tau \left({n}\right) \\ \hline 1 & 1 \\ 2 & 2 \\ 3 & 2 \\ 4 & 3 \\ 5 & 2 \\ 6 & 4 \\ 7 & 2 \\ 8 & 4 \\ 9 & 3 \\ 10 & 4 \\ 11 & 2 \\ 12 & 6 \\ 13 & 2 \\ 14 & 4 \\ 15 & 4 \\ 16 & 5 \\ \end{array}$

This sequence is A000005 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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