Dickson (1913, 1952) defined an amicable triple to be a Triple of three numbers such that

where is the Restricted Divisor Function (Madachy 1979). Dickson (1913, 1952) found eight sets of amicable triples with two equal numbers, and two sets with distinct numbers. The latter are (123228768, 103340640, 124015008), for which

and (1945330728960, 2324196638720, 2615631953920), for which

A second definition (Guy 1994) defines an amicable triple as a Triple such that

where is the Divisor Function. An example is ( , , ).

**References**

Dickson, L. E. ``Amicable Number Triples.'' *Amer. Math. Monthly* **20**, 84-92, 1913.

Dickson, L. E. *History of the Theory of Numbers, Vol. 1: Divisibility and Primality.* New York: Chelsea, p. 50, 1952.

Guy, R. K. *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag,
p. 59, 1994.

Madachy, J. S. *Madachy's Mathematical Recreations.* New York: Dover, p. 156, 1979.

Mason, T. E. ``On Amicable Numbers and Their Generalizations.'' *Amer. Math. Monthly* **28**, 195-200, 1921.

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

© 1996-9

1999-05-25