# sociable numbers

Recall that we form an aliquot sequence by taking a positive integer, adding all of its positive divisors other than itself, and repeating the process with this sum. For example, starting with 20 we get 20, 22 (1+2+4+5+10), 14 (1 +2 + 11), 10 (1 + 2 + 7), 8, 7, 1, 0, 0, . . . Sometimes this process results in a cycle (which repeats indefinitely) such as

14288, 15472, 14536, 14264, 12496, 14288, 15472, 14536, 14264, 12496, . . .

These repeating numbers are called sociable numbers (and if they have length two, amicable numbers).

In 1918 Poulet found the example above of length five and the following chain of length 28.

14316, 19116, 31704, 47616, 83328, 177792, 295488, 629072, 589786, 294896, 358336, 418904, 366556, 274924, 275444, 243760, 376736, 381028, 285778, 152990, 122410, 97946, 48976, 45946, 22976, 22744, 19916, 17716

In 1969 Borho [BH1986] constructed one of length 4:

28158165, 29902635, 30853845, 29971755

Before involving electronic computers in the search, only three chains of sociable numbers were known. Now over 5000 such chains are known (mostly of length four).

See Also: AliquotSequence, SigmaFunction

References:

BH1986
Borho, W. and Hoffmann, H., "Breeding amicable numbers in abundance," Math. Comp., 46:173 (1986) 281--293.  MR 87c:11003
Cohen70
H. Cohen, "On amicable and sociable numbers," Math. Comp.,:24 (1970) 423-429.
RT2015
Rocha, Rodrigo and Thatte, Bhalchandra, Distributed cycle detection in large-scale sparse graphs.  October 2015. (Abstract available)