Suppose we define the function as the sum of the divisors of n
Express in terms of ‘s prime factorization
Let’s just look at a quick example to help us out.
Suppose that we wanted to find the sum of the divisors of . We enumerate these in the following order: first, we decrease the exponent on the 5, then that of 4, and finally 3, like we’re counting in base .
Okay, but what about the next 4 terms?
Now, without any loss of generality, the next 12 terms will all be the same, and we’ll end up with , at which point we just decrease the exponent on the two, and the same exact operations, again without loss of generality, will eventually lead to
Using the same argument, we can justify that for some arbitrary factorized as , then its sum will just be
A lot of you may recognize that every one of the clauses are all geometric series, hence we can simplify this expression as