sigma function

The sigma function of a positive integer n is the sum of the positive divisors of n. This is usually σ(n) using the greek letter sigma.

Here are the first few values of this function:

integer n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
σ(n)	1	3	4	7	6	12	8	15	13	18	12	28	14	24	24	31

Clearly, for primes p, σ(p)=p+1. The function σ(x) is a multiplicative function, so its value can be determined from its value at the prime powers:

Theorem: If p is prime and n is any positive integer, then σ(pⁿ) is (pⁿ⁺¹-1)/(p-1).

Example:

σ(2000) = σ(2⁴5³) = σ(2⁴)^.σ(5³) = (2⁵-1)/(2-1) ^. (5⁴-1)/(5-1) = 31 ^. 156 = 4836.