[Math Lair] Sociable Numbers

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Sociable numbers are similar to amicable numbers. A chain of numbers is sociable if the sum of the proper divisors of each number is the next number in the chain, the last number preceding the first. The first two chains were found by Poulet in 1918. The first chain contains five members, 12,496 → 14,288 → 15,472 → 14,536 → 14,264 → 12,496, while the second chain contains a remarkable 28 numbers.

These two chains were the only known sociable chains until 1969, when Henri Cohen used a computer to check all numbers below 60,000,000, and he discovered seven new chains of four links. More have been found since then.

Perfect numbers and amicable numbers could be considered special cases of sociable numbers. A perfect number could be considered as a chain of length 1, while amicable numbers could be seen as a chain of length 2. Curiously, no chains with just three links (which someone named "crowds") have been found.

This process of traversing sociable chains is somewhat reminiscent of looping. See also aliquot chains.