# 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 .

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

is then

The average order of σ(*n*) is .

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*) = 2*n*. A number is similarly defined to be *abundant* if σ(*n*) > 2*n* and *deficient* if σ(*n*) < 2*n*. A pair of numbers *m*, *n* are *amicable* if σ(*m*) = *m*+*n* = σ(*n*): the smallest such pair is 220 and 284.