Amicable Numbers

Amicable numbers are related, in some sense, to perfect numbers. A perfect number, if you recall, is a number whose divisors (not including itself) add up to itself. An amicable pair is two different numbers such that the sum of the divisors of one of them (not including the number itself) adds up to the other number; and the sum of the divisors of the other number (not including that number itself) adds up to the first number. The numbers 220 and 284 are such numbers. If you find the divisors of 220 and sum them (not including 220) you will get 284. And if you find the divisors of 284 and sum them (not including 284) you will get 220. Thus the numbers are amicable (or friendly).

There are chains of such numbers as well, where the sum of the divisors of the first equal the second; the sum of the divisors of the second equal the third; and so on until the sum of the divisors of the last number in the list equals the first number in the list. These kinds of numbers are called sociable.

Here are some pairs to verify:

• (220, 284),
• (1184, 1210),
• (2620, 2924),
• (5020, 5564),
• (6232, 6368)

And 12496 is the first number in a 5-number sociable set of numbers. Can you generate the rest of the chain?