## Aliquot Sequence

Let

where is the Divisor Function and is the Restricted Divisor Function. Then the Sequence of numbers

is called an aliquot sequence. If the Sequence for a given is bounded, it either ends at or becomes periodic.
1. If the Sequence reaches a constant, the constant is known as a Perfect Number.

2. If the Sequence reaches an alternating pair, it is called an Amicable Pair.

3. If, after iterations, the Sequence yields a cycle of minimum length of the form , , ..., , then these numbers form a group of Sociable Numbers of order .
It has not been proven that all aliquot sequences eventually terminate and become period. The smallest number whose fate is not known is 276, which has been computed up to (Guy 1994).

See also 196-Algorithm, Additive Persistence, Amicable Numbers, Multiamicable Numbers, Multiperfect Number, Multiplicative Persistence, Perfect Number, Sociable Numbers, Unitary Aliquot Sequence

References

Guy, R. K. Aliquot Sequences.'' §B6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 60-62, 1994.

Guy, R. K. and Selfridge, J. L. What Drives Aliquot Sequences.'' Math. Comput. 29, 101-107, 1975.

Sloane, N. J. A. Sequence A003023/M0062 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and extended entry in Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.