Several of the categories of numbers defined by the ancient Greeks (including perfect, deficient, and abundant numbers) depend on the sum of the positive divisors of n excluding itself. We denote this divisor sum using the sigma function σ(n). When we compare n with σ(n)-n, there are just three possibilities:
possibility n is then called a σ(n)-n < n deficient number σ(n)-n = n perfect number σ(n)-n > n abundant number
A common question for a modern mathematician to ask is "what if we iterate this function?" For example, starting with 20, we would 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=1, then 0, then 0 again (zero will now repeat forever). It is traditional to stop this process when we reach one.
These iterated sequences are called aliquot sequence. Here is another: 12, 16, 15, 9, 4, 3, 1. Do they always end in one? No! (Why don't you try starting with a perfect number such as 6 or 28?)
Notice that if we ever repeat a number, then we are caught in a loop and will continue to repeat. For example 220, 284, 220, 284, . . . (a pair of amicable numbers); or
14288, 15472, 14536, 14264, 12496, 14288, 15472, 14536, 14264, 12496, . . .
These sets of repeating numbers are called sociable chains and also aliquot cycles.
So do these aliquot sequences always either end in one or in an aliquot cycle? This is another open question! In 1888, M. E. Catalan conjectured that they do, but others, like Guy and Selfridge, suggest this may be another case of the law of small numbers. In fact, some think that almost all of those aliquot sequences that start at an even number will never repeat.
Here are the numbers below 2000 for which it is unknown if an aliquot sequence containing them will end in one or repeat (all the others have been checked):
276, 552, 564, 660, 966, 1074, 1134, 1464, 1476, 1488, 1512, 1560, 1578, 1632, 1734, 1920, 1992
See the pages linked below for more information on the status of aliquot sequences.
Related pages (outside of this work)
- aliquot sequences by Paul Zimmermann
- aliquot sequences by Juan L. Varona
- aliquot sequences by Wolfgang Creyaufmueller