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From: cscussel@ihuxq.UUCP (Chris Scussel)
Newsgroups: net.math
Subject: Another number theoretic recursion: s(n)
Message-ID: <359@ihuxq.UUCP>
Date: Tue, 25-Oct-83 00:26:27 EDT
Article-I.D.: ihuxq.359
Posted: Tue Oct 25 00:26:27 1983
Date-Received: Thu, 20-Oct-83 04:54:42 EDT
Organization: AT&T Bell Labs, Naperville, Il
Lines: 25

Here's another interesting function to iterate. Consider s(n), the sum of the
divisors of n (except n). For example, s(6)=6, s(4)=1+2=3. Now, consider
s(n), s(s(n)), s(s(s(n))). This sequence can cycle (with period 1 for perfect
numbers, 2 for amicable pairs, etc), collapse to 1 (another form of cycling)
or fail to cycle (and thus increase without bound, although not necessarily
monotonically). I've seen books that say failure to cycle is possible, and
others that say cycling must occur (possibly after an initial finite noncyclic
sequence). This is easy to try on your favorite machine (although not quite
as easy as the triple-or-half problem) and can eat up some CPU time (138 is
a nice number to try, if you have the time). Of course, you can't find a number
that gives an unbounded sequence by checking out the sequence on a computer.
But, it's fun to look...

By the way, it's easy to evaluate s(n) by just adding all the divisors you can
find that are less then n. But it's lots faster to only go up to n/2. Unless
you only go up to the square root, though, you'll take forever to do 138.
The trick is if d divides n, add d and n/d to the sum, unless d=n/d.

Anyone interested???

					Chris Scussel
					Bell Labs
					Naperville, Illinois

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