Definition:Sociable Chain

Definition
Let $m$ be a positive integer.

Let $\map s m$ be the aliquot sum of $m$.

Define the sequence $\sequence {a_k}$ recursively as:
 * $a_{k + 1} = \begin{cases} m & : k = 0 \\ \map s {a_k} & : k > 0 \end{cases}$

A sociable chain is such a sequence $\sequence {a_k}$ where:
 * $a_r = a_0$

for some $r > 0$.

Also known as
A sociable chain is known in some sources as a sociable cycle.

Beware of the temptation to refer to it as a social chain -- this term is not used in mathematics.

A sociable chain of order $2$ is generally known as an amicable pair.

A sociable chain of order $3$ is also known by some mathematicians as a crowd, but it is unknown whether any exist.

Also see

 * Definition:Sociable Number
 * Definition:Aliquot Sequence


 * Existence of Sociable Chain of Order 3