Sociable Numbers — Definition, Formula & Examples

Sociable numbers are a set of numbers that form a cycle when you repeatedly compute the sum of proper divisors: starting from any number in the set, the chain of divisor sums eventually returns to that starting number.

A sociable chain of order $k$ is a cyclic sequence of distinct positive integers $n_1, n_2, \ldots, n_k$ such that $s(n_i) = n_{i+1}$ for $1 \le i < k$ and $s(n_k) = n_1$ , where $s(n)$ denotes the aliquot sum (the sum of all proper divisors of $n$ ). Perfect numbers are sociable chains of order 1, and amicable pairs are sociable chains of order 2.

Key Formula

s(n) = \sum_{\substack{d \mid n \\ d < n}} d

Where:

$s(n)$ = The aliquot sum of n — the sum of all proper divisors of n
$d \mid n$ = d is a positive divisor of n

How It Works

To check whether a number belongs to a sociable chain, compute its aliquot sum (the sum of its proper divisors, excluding the number itself). Then compute the aliquot sum of that result, and repeat. If you return to the original number after

k

steps, those

k

numbers form a sociable chain of order

k

. If the chain never returns, the number is not sociable.

Worked Example

Problem: Verify that 12496, 14288, 15472, 14536, 14264 form a sociable chain of order 5.

Step 1: Compute the sum of proper divisors of 12496.

s(12496) = 14288

Step 2: Continue the chain by computing each successive aliquot sum.

s(14288) = 15472,\quad s(15472) = 14536,\quad s(14536) = 14264

Step 3: Compute the aliquot sum of the last number to check if the cycle closes.

s(14264) = 12496

Answer: The chain returns to 12496 after 5 steps, confirming these five numbers form a sociable chain of order 5. This was the first sociable chain discovered (by Poulet in 1918).

Why It Matters

Sociable numbers connect to deep open questions in number theory, including the Catalan–Dickson conjecture about whether all aliquot sequences eventually terminate or cycle. Studying them strengthens skills in divisor functions and factoring that appear throughout number theory courses and mathematical competitions.

Common Mistakes

Mistake: Including the number itself when computing the aliquot sum.

Correction: The aliquot sum uses only proper divisors — all divisors strictly less than the number. Including the number itself gives the full divisor sum σ(n), which breaks the cycle check.