Sum-of-divisors function

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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

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

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.