How to find "sum of all factors" of composite no. N?

Question

any general formula?

All i know is that "product of all factors" of N = (a^b) x (c^d) x ... = a^(1+2+...b) x c^(1+2+...d) x ....

Any similar thing to find "sum of all factors"?
Thank you.

Answer 1

In general : where p, q, r ... are the factors
and a, b, c ... are the number of each factor :

N = (p^a) * (q^b) * (r^c) * ...

Sum of factors =
[p^(a+1) - 1] / (p-1) * [q^(b+1) - 1] / (q-1) * [r^(c+1) - 1] / (r-1) *...

Example :

18000 = 2^4 * 3^2 * 5^3

Sum of all factors
= [2^(4+1) - 1] / (2-1) * [3^(2+1) - 1] / (3-1) * [5^(3+1) - 1] / (5-1)

= 31 * 13 * 156

= 62868

Answer 2

The sum of the factors of a^b is a + a^2 + ... + a^b =
[a^(b+1) - 1]/[a - 1] if a NE 0, and = b if a = 1. Add in the other sums and you have your answer.