AliquotSequence 2006-07-28 16:49:50 2006-08-09 16:46:53 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 For a given $m$, define the recurrence relation $a_1 = m$, $a_n = \sigma(a_{n - 1}) - a_{n - 1}$, where $\sigma(x)$ is the sum of divisors function. $a$ is then the {\em aliquot sequence} of $m$. If $m$ is an amicable number, its aliquot sequence is periodic, alternating between the abundant and deficient member of the amicable pair. For a prime number $p$, its aliquot sequence is $p, 1, 0$. In other cases, the aliquot sequence reaches a fixed point upon 0, or on a perfect number.