[Math Lair] Aliquot Chains

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Aliquot chains, which are sometimes referred to as sociable chains, are iterative processes or loops. The process is:

  1. Start with any number.
  2. Find the sum of the number's proper divisors (also known as aliquot parts).
  3. Take the sum of the proper divisors, and find its sum of proper divisors, and so on and so forth.

The behaviour of these loops varies. Most chains end at zero (if you have a prime number, its only proper divisor is 1, and 1 has no proper divisors). A perfect number will sum to itself again and again and again. Amicable or sociable numbers will iterate over the same set of two, or four, or even more, numbers. There are a few numbers that seem to increase indefinitely.