Sum-of-divisors function

In number theory the sum-of-divisors function of a positive integer, denoted σ(n), is the sum of the positive divisors of the number n.

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

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

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

is then

$\sigma (n)=\prod _{i}\left(1+p+p^{2}+\cdots +p_{i}^{a_{i}}\right).\,$

The average order of σ(n) is ${\frac {\pi ^{2}}{6}}n$.