An aliquot part, also known as a proper divisor, is any divisor of a number that isn't equal to the number itself. For instance, the aliquot parts of 12 are 1, 2, 3, 4, and 6. The word comes from the Latin ali ("other") and quot ("how many"). An aliquot sequence is formed by taking the sum of the aliquot parts of a number, adding them to form a new number, then repeating this process on the next number and so on. For example, starting with 20, we get 1 + 2 + 4 + 5 + 10 = 22, then 1 + 2 + 11 = 14, then 1 + 2 + 7 = 10, then 1 + 2 + 5 = 8, then 1 + 2 + 4 = 7, then 1, after which the sequence doesn't change. For some numbers, the result loops back immediately to the original number; in such cases the two numbers are called amicable numbers. In other cases, where a sequence repeats a pattern after more than one step, the result is known as an aliquot cycle or a sociable chain. An example of this is the sequence 12496, 14288, 15472, 14536, 14264, ... The aliquot parts of 14264 add to give 12496, so that the whole cycle begins again.

 

Do all aliquot sequences end either in 1 or in an aliquot cycle (of which amicable numbers are a special case)? In 1888, the Belgian mathematician Eugène Catalan (1814–1894) conjectured that they do, but this remains an open question.