# aliquot part

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 EugĂ¨ne Catalan (1814–1894) conjectured that they do, but this remains an open question.