# Aliquot Sequences, Amicable Pairs, and Sociable Numbers

Recall from The Sum of Positive Divisors of an Integer page that if $n \in \mathbb{Z}$ then $\sigma(n)$ denotes the sum of all positive divisors of $n$ and is explicitly defined as:

(1)We will now look at a special type of sequence involving the function $\sigma$.

Definition: The Aliquot Sequence of $n \in \mathbb{N}$ is the sequence whose terms are defined recursively by $s_0 = n$ and $s_k = \sigma(s_{k-1}) - s_{k-1}$ for each $k \in \mathbb{N}$. |

For example, consider $n = 14$. The corresponding aliquot sequence is as follows:

(2)So the aliquot sequence of $14$ is $(14, 10, 8, 7, 1, 0)$

One important thing to note is that if $n$ is a perfect number, then $\sigma(n) = 2n$, and so:

(3)Moreover, if $s_j = s_k$ for some $j, k \in \mathbb{N}$ then the aliquot sequence of $n$ will repeat periodically.

So we would like to know whether there exists an $n$ such that the aliquot sequence does not terminate trivially with either $0$, a perfect number, or periodically repeats. This is a famous unanswered question and there are certain candidate numbers such as $276$ which may fit this criterion.

Definition: Two numbers $m$ and $n$ are said to be an Amicable Pair if $\sigma(m) - m = n$ and $\sigma(n) - n = m$. |

Equivalently, $m$ and $n$ are an amicable pair if and only if:

(4)The smallest amicable pair of $(220, 284)$. You should indeed verify that this is an amicable pair.

Definition: A collection of numbers $\{ n_1, n_2, ..., n_k \}$ are said to be Sociable if $\sigma(n_1) - n_1 = n_2$, $\sigma(n_2) - n_2 = n_3$, …, $\sigma(n_k) - n_k = n_1$. |

For example, the following set of numbers is sociable as you should verify:

(5)