Number of divisors function

In number theory the number of divisors function of a positive integer, denoted d(n) or τ(n), is the number of positive integer divisors of the number n.

It is a multiplicative function, that is is m and n are coprime then $d(mn)=d(m)d(n)$.

The value of d on a general integer n with prime factorisation

$n=\prod _{i}p_{i}^{a_{i}}\,$

is then

$d(n)=\prod _{i}\left(a_{i}+1\right).\,$

The average order of d(n) is $\log(n)$.