is defined as the sum of positive divisors of , including and . This can be written as
Properties
Theorem
Proof:
Let then the divisors will have the form . Every possible divisor appears exactly once so each combination of will appear exactly once which can be written as which are geometric series, therefore
Let . Every positive divisor .
Therefore, the sum of all positive divisors of is given by
This multiple summation can be factored into a product of separate geometric series:
Now suppose and are two positive integers with , and let and where the and are distinct primes (since and are coprime). Then , and using the previous identity for , we get