Let
 |
where 
is the divisor function and 
is the restricted
divisor function. Then the sequence of numbers
 |
is called an aliquot sequence. If the sequence for a given 
is bounded, it either ends at 
or becomes periodic.
1. If the sequence is a constant, the constant is known as a perfect number. A number that is not perfect, but for which the sequence becomes constant, is known as an aspiring number.
2. If the sequence reaches an alternating pair, it is called an amicable pair.
3. If, after 
iterations, the sequence yields a cycle of minimum length
of the form 
, 
, ..., 
, then these numbers form a group of sociable
numbers of order 
.
The lengths of the aliquot sequences for 
, 2, ... are 1, 2, 2, 3, 2, 1, 2, 3, 4, 4, 2, 7, 2, 5, 5,
6, 2, ... (OEIS A044050).
It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. Guy (1994) cites the
largest computed value as 
, though this has since been extended to 
(Zimmermann 2008). There are five such sequences
less than 1000, namely 276, 552, 564, 660, and 966 (Clavier 2006, Varona 2004), sometimes
called the "Lehmer five" (Zimmermann 2008). Furthermore, there are 81 open
sequences 
,
908 open sequences 
,
and 9452 open sequences 
(Creyaufmüller 2008).
