Aliquot Sequence

Let

$$s(n) \equiv \sigma (n)-n,$$

where $\sigma(n)$ is the Divisor Function and $s(n)$ is the Restricted Divisor Function. Then the Sequence of numbers

$$s^0(n)\equiv n, s^1(n) = s(n), s^2(n) = s(s(n)), \ldots$$

is called an aliquot sequence. If the Sequence for a given $n$ is bounded, it either ends at $s(1)=0$ 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 $k$ iterations, the Sequence yields a cycle of minimum length $t$ of the form $s^{k+1}(n)$, $s^{k+2}(n)$, ..., $s^{k+t}(n)$, then these numbers form a group of Sociable Numbers of order $t$.

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 $s^{628}(276)$ (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.