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 divisor
s. In simple terms, each term in the sequence is the sum of the previous term's proper divisor
s (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.