sociable numbers
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000: 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 [BOR1969] 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 fifty such chains are known!

See Also: AliquotSequence, SigmaFunction


Chris K. Caldwell © 1999-2017 (all rights reserved)