sigma function
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: The sigma function of a positive integer n is the sum of the positive divisors of n. This is usually denoted sigma(n) 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 123456 78 910111213 14 1516
sigma(n) 1347 612 815 131812281424 2431

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.




Chris K. Caldwell © 1999-2016 (all rights reserved)