Newsgroups: sci.math.research
From: hoey@aic.nrl.navy.mil (Dan Hoey)
Date: 1995/11/18
Subject: Re: Sociable Chains

r...@ix.netcom.com (Robert G. Wilson v)

> The function, Sum of the proper Divisors, or the Sum of the
> Aliquiont parts of X, is a well known Number Theory arithmetic
> function...

Guy, in _Unsolved Problems in Number Theory, 2nd edition_[1994],
calls the cycles "aliquot cycles".  (I've never seen the spelling
"aliquiont", though my dictionary says "aliquant" is a word for a
non-divisor!?  Mercifully, they both seem to be obsolete except in
this usage).  Perfect numbers, sociable pairs, aliquot cycles.

> I understand that a sociable of cycle 3 exists....

Unless you've heard very recently, no cycles of length 3, 7, or >9
are known, save for one cycle of length 28 beginning at 14316, found
by Poulet.  "It has been conjectured that there are no 3-cycles.
On the other hand it has been conjectured that for each k there
are infinitely many k-cycles."

Dan Hoey                                [posted and emailed]
Hoey@AIC.NRL.Navy.Mil