[seqfan] Re: Destinies of sums of proper divisors.

franktaw at netscape.net franktaw at netscape.net
Mon Jul 6 19:48:26 CEST 2009

Actually, if n is large with many prime divisors, and is divisible by 
larger power of 2, the next term is quite likely to continue to be 
divisible by that power of 2; it is perhaps as likely to to increase 
this exponent as it is to decrease it.  So these trajectories do not, 
in general, get stuck in this particular pattern.

On the other hand, almost everybody who has worked with these things 
believes that there are, indeed, divergent trajectories, and that 276 
is very likely one of them.

Franklin T. Adams-Watters

-----Original Message-----
From: David Wilson <dwilson at gambitcomm.com>

I also wanted to make this observation on aliquot sequences:

Let s(n) be the sum of divisors of n, and f(n) = s(n)-n be the sum of
aliquot divisors. We are discussing the trajectory of f(n).

Note that if n is a large number of the form 28k with k odd, then f(n)
is likely to be a yet larger number of this form. If a sufficiently
large number of this form occurs in the trajectory of n, the subsequent
trajectory is likely retain that form and grow. Once it has grown
sufficiently large, in the unlikely event it loses the form 28k, it is
likely to recover that form after a few iterations
and resume its growth.

For this reason, I conjecture that f(n) is very likely to include
divergent trajectories, and that n = 276 is probably one of them.

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