Sociable Numbers

Numbers which result in a periodic Aliquot Sequence. If the period is 1, the number is called a Perfect Number. If the period is 2, the two numbers are called an Amicable Pair. If the period is $t\geq 3$, the number is called sociable of order $t$. Only two sociable numbers were known prior to 1970, the sets of orders 5 and 28 discovered by Poulet (1918). In 1970, Cohen discovered nine groups of order 4.

The table below summarizes the number of sociable cycles known as given in the compilation by Moews (1995).

Order	Known
3	0
4	38
5	1
6	2
8	2
9	1
28	1

See also Aliquot Sequence, Perfect Number, Unitary Sociable Numbers

References

Beeler, M.; Gosper, R. W.; and Schroeppel, R. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239, Item 61, Feb. 1972.

Borho, W. ``Über die Fixpunkte der $k$-fach iterierten Teilerersummenfunktion.'' Mitt. Math. Gesellsch. Hamburg 9, 34-48, 1969.

Cohen, H. ``On Amicable and Sociable Numbers.'' Math. Comput. 24, 423-429, 1970.

Devitt, J. S.; Guy, R. K.; and Selfridge, J. L. Third Report on Aliquot Sequences, Congr. Numer. XVIII, Proc. 6th Manitoba Conf. Numerical Math, pp. 177-204, 1976.

Flammenkamp, A. ``New Sociable Numbers.'' Math. Comput. 56, 871-873, 1991.

Gardner, M. ``Perfect, Amicable, Sociable.'' Ch. 12 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 160-171, 1978.

Guy, R. K. ``Aliquot Cycles or Sociable Numbers.'' §B7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 62-63, 1994.

Madachy, J. S. Madachy's Mathematical Recreations. New York: Dover, pp. 145-146, 1979.

Moews, D. and Moews, P. C. ``A Search for Aliquot Cycles Below $10^{10}$.'' Math. Comput. 57, 849-855, 1991.

Moews, D. and Moews, P. C. ``A Search for Aliquot Cycles and Amicable Pairs.'' Math. Comput. 61, 935-938, 1993.

Moews, D. ``A List of Aliquot Cycles of Length Greater than 2.'' Rev. Dec. 18, 1995. http://xraysgi.ims.uconn.edu:8080/sociable.txt.

Poulet, P. Question 4865. L'interméd. des Math. 25, 100-101, 1918.

te Riele, H. J. J. ``Perfect Numbers and Aliquot Sequences.'' In Computational Methods in Number Theory, Part I. (Eds. H. W. Lenstra Jr. and R. Tijdeman). Amsterdam, Netherlands: Mathematisch Centrum, pp. 141-157, 1982.

Weisstein, E. W. ``Sociable and Amicable Numbers.'' Mathematica notebook Sociable.m.