[Math Lair] Aliquot Chains

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Aliquot chains, which are sometimes referred to as sociable chains or aliquot sequences, 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. Whenever a chain reaches a prime number, the next number will be 1, because the only proper divisor of a prime is 1; 1 has no proper divisors, so the next term will be 0. Some chains can take a while to get there, though. 138 takes 178 links to get to 1. 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. The smallest number that I've found with this behaviour is 276. It is not known whether the sequence 276, or similar numbers, continues forever or eventually returns to 1.

You can see the behaviour of all numbers under 1,000 on aliquot chains for all numbers less than 1,000.