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Given a natural number n, the aliquot sequence beginning with n is defined by f(1)=n and f(i+1)=sigma(f(i))-f(i), where sigma(n) is defined as the sum of all n's positive divisors. In simple terms, each term in the sequence is the sum of the previous term's proper divisors (those positive divisors which do not equal the number itself).

As of this writing, nobody knows exactly how aliquot sequences behave. Obviously if a certain sequence hits a prime, then the next term will be 1 and the sequence terminates; another possibility is that the sequence will end up in a repeating loop caused by the appearance of a perfect number, amicable pair or sociable chain. However, it is not known whether an infinite non-repeating aliquot sequence exists, and in fact some of the contenders for this title begin with numbers as small as three digits.

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