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 d(mn)=d(m)d(n).

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

n=∏ipiai

is then

d(n)=∏i(ai+1).

The average order of d(n) is log⁡(n). The normal order of log(d(n)) is log(2) log log(n).