Sum-Of-Divisors Function

	Main Article	Discussion	Related Articles  [?]	Bibliography  [?]	External Links  [?]	Citable Version  [?]
	This editable Main Article is under development and subject to a disclaimer. [edit intro]

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

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

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

${\displaystyle n=\prod _i{p}_i^{a_i}}$

is then

${\displaystyle \sigma (n)=\prod _i(1+p+p^2+\cdots +p_i^{a_i}).}$

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

A perfect number is defined as one equal to the sum of its "aliquot divisors", that is all divisors except the number itself. Hence a number n is perfect if σ(n) = 2n. A number is similarly defined to be abundant if σ(n) > 2n and deficient if σ(n) < 2n. A pair of numbers m, n are amicable if σ(m) = m+n = σ(n): the smallest such pair is 220 and 284.