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 .

**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.

© 1996-9

1999-05-25