Aliquot Sequences, Amicable Pairs, and Sociable Numbers

# 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)
\begin{align} \quad \sigma(n) = \sum_{d \mid n}_{d > 0} d \end{align}

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)
\begin{align} \quad s_0 & = 14 \\ \quad s_1 & = \sigma(14) - 14 = \sigma(2) \sigma(7) - 14 = (1 + 2)(1 + 7) - 14 = 24 - 14 = 10 \\ \quad s_2 & = \sigma(10) - 10 = \sigma(2) \sigma(5) - 10 = (1 + 2)(5 + 1) - 10 = 18 - 10 = 8 \\ \quad s_3 & = \sigma(8) - 8 = \sigma(2^3) - 8 = (1 + 2 + 4 + 8) - 8 = 7 \\ \quad s_4 & = \sigma(7) - 7 = (1 + 7) - 7 = 1 \\ \quad s_5 &= \sigma(1) - 1 = 1 - 1 = 0 \\ \quad s_k &= \sigma(0) - 0 = 0 - 0 = 0 \quad \forall k \geq 6 \end{align}

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)
\begin{align} \quad s_0 & = n \\ \quad s_1 & = \sigma(s_0) - s_0 = 2n - n = n \\ \quad s_2 & = n \\ \quad & \vdots \end{align}

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)
\begin{align} \quad \sigma(m) = \sigma(n) \end{align}

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)
\begin{align} \quad \{ 1264460, 1547860, 1727636, 1305184 \} \end{align}