Aliquot sequence

starting from

The sequence of natural numbers defined by the rule

The sequence is said to be terminating if for some and eventually periodic if there is a such that for sufficiently large. If , then is a perfect number, while if , then and form an amicable pair (cf. also Amicable numbers).

An example of an eventually periodic aliquot sequence is the sequence . Larger cycles are possible; e.g., a sequence with cycle length is known.

The Catalan–Dickson conjecture states that all aliquot sequences either terminate or are eventually periodic. This conjecture is still (1996) open, but generally thought to be false.

References

[a1]	H.J.J. te Riele, "A theoretical and computational study of generalized aliquot sequences" , Math. Centre , Amsterdam (1976)