The algebraic independence of the sum of divisors functions

Let σj(n)=∑d|ndj be the sum of divisors function, and let I be the identity function. When considering only one input variable n, we show that the set of functions {σi}i=0∞∪{I} is algebraically independent. With two input variables, we give a non-trivial identity involving the sum of divisors function, prove its uniqueness, and use it to prove that any perfect number n must have the form n=rσ(r)/(2r−σ(r)), with some restrictions on r. This generalizes the known forms for both even and odd perfect numbers.