# Aliquot sequence

2010 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

starting from $n$

The sequence of natural numbers $a_1,a_2,\dots$ defined by the rule $a_1 = n$, $a_{k+1} = s(a_k)$ where $s(a)$ is the sum of aliquot divisors function $$s(a) = \sum_{d|a}d - a \ .$$

The sequence is said to be terminating if $a_n=1$ for some $n$ and eventually periodic if there is a $c$ such that $a_{n+c}=a_n$ for all $n$ sufficiently large. If $a_{n+1}=a_n$, then $a_n$ is a perfect number, while if $a_{n+2}=a_n$, then $a_n$ and $a_{n+1}$ form an amicable pair (cf. also Amicable numbers): aliquot cycles of length greater than 2 are also termed sociable numbers.

An example of an eventually periodic aliquot sequence is the sequence $562,220,284,220,\dots$. Longer cycles are known; e.g., a sequence with cycle length $28$, starting at $n=14316$ (ref [b2]).

The Catalan–Dickson conjecture states that all aliquot sequences either terminate or are eventually periodic. This conjecture is still (1996) open, but generally thought to be false. The aliquot sequence starting at $n = 3556$ is of length $2058$ (ref [b1]).

How to Cite This Entry:
Aliquot sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Aliquot_sequence&oldid=36089
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article