SumOfDivisorsFunction 2006-07-28 16:29:44 2009-01-19 02:44:07 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a positive integer $n$, the sum of the integers $0 < d \le n$ such that $d|n$ is the value of the {\em sum of divisors function} for $n$, often symbolized by a Greek lowercase $\sigma$. Thus, $$\sigma(n) = \sum_{d|n} d.$$ Sometimes this function is referred to as $\sigma_1(n)$, highlighting its relation to the divisor function. Given coprime integers $m$ and $n$ (that is, $\gcd(m, n) = 1$) then $\sigma(mn) = \sigma(m)\sigma(n)$, meaning that the sum of divisors function is a multiplicative function.