Aliquot Sequences

Aliquot Sequences

Definition: The aliquot sequence for a number n is defined recursively with the first element in the sequence being $s_0 = n$, and the kth element being $s_k = \sigma (s_{k-1}) - s_{k-1}$.

The explanation of what an aliquot sequence specifically is, is best explained with an example:

Example 1

Find the first 5 terms of the aliquot sequence for 24.

The first term in our aliquot sequence, $s_0 = 24$.

  • The next term in our aliquot sequence is:
(1)
\begin{align} s_1 = \sigma (24) - 24 \\ s_1 = \sigma (2^3) \sigma (3) - 24 \\ s_1 = (15)(4) - 24 \\ s_1 = 36 \end{align}
  • The next term in our aliquot sequence is:
(2)
\begin{align} s_2 = \sigma (36) - 36 \\ s_2 = \sigma (2^2) \sigma (3^2) - 36 \\ s_2 = (7)(13) - 36 \\ s_2 = 55 \end{align}
  • The next term in our aliquot sequence is:
(3)
\begin{align} s_3 = \sigma (55) - 55 \\ s_3 = \sigma(5) \sigma (11) - 55 \\ s_3 = (6)(12) - 55 \\ s_3 = 17 \end{align}
  • The next term in our aliquot sequence is:
(4)
\begin{align} s_4 = \sigma (17) - 17 s_4 = 18 - 17 \\ s_4 = 1 \end{align}
  • The next term in our aliquot sequence is:
(5)
\begin{align} s_5 = \sigma (1) - 1 \\ s_5 = 1 - 1 \\ s_5 = 0 \end{align}

So our aliquot sequence has terminated. The full sequence is (24, 36, 55, 17, 1, 0).

Perfect Numbers, Amicable Pairs, and Sociable Numbers

Perfect Numbers

Recall that a perfect number n satisfies $\sigma (n) = 2n$.

If we try to take the aliquot sequence of a perfect number n, we will find that the sequence will be (n, n, n, …). This is because $\sigma (n) - n = n$ for all perfect numbers, since this is just a rewriting of $\sigma (n) = 2n$. For example, if we try to take the aliquot sequence of 6, our next term in the sequence will be $s_1 = \sigma (6) - 6 = 6$, and our next term in the sequence will be $s_2 = \sigma (6) - 6 = 6$, and so forth.

Amicable Pairs

Definition: Two positive integers m and n are said to be an amicable pair if and only if $\sigma (m) - m = n$ and $\sigma (n) - n = m$ forming the aliquot sequence $(x_1, x_2, x_1, x_2, x_1, ...)$.

In the case of amicable pairs starting with integer m, we will find that their aliquot sequence will be (m, n, m, n, …).

Sociable Numbers

Definition: For n > 2, the set of n positive integers $x_1, x_2, ..., x_n$ are said to be sociable numbers if and only if $\sigma(x_1) - x_1 = x_2$, $\sigma (x_2) - x_2 = x_3$, …, $\sigma (x_{n-1}) - x_{n-1} = x_n$, and $\sigma (x_n) - x_n = x_1$ forming the aliquot sequence $(x_1, x_2, ..., x_{n-1}, x_n, x_1, x_2, ...)$.

A common example of sociable number is 1264460 that has an aliquot sequence of (1264460, 1547860, 1727636, 1305184, 1264460, 1547860, …) Notice that when n = 1, the aliquot sequence defines a perfect number, and when n = 2, the aliquot sequence defines an amicable pair. So the cycle formed by sociable numbers must be greater than 2 by definition.

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