The Prime Glossary: sigma function

sigma function
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The sigma function of a positive integer n is the sum of the positive divisors of n. This is usually denoted

using the greek letter sigma

, but for those with non-graphical browsers we will use sigma(n) on these pages. Here are the first few values of the sigma function:

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

Clearly, for primes p, sigma(p)=p+1. sigma(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 sigma(pⁿ) is (pⁿ⁺¹-1)/(p-1).

Example:

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