Number of divisors function

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In number theory the number of divisors function of a positive integer, denoted d(n) or τ(n) or σ₀(n), is the number of positive integer divisors of the number n.

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

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

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

is then

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

The average order of d(n) is ${\displaystyle \log(n)}$. The normal order of log(d(n)) is log(2) log log(n).