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

David Wilson dwilson at gambitcomm.com
Mon Jul 6 15:25:28 CEST 2009

I believe the proper terms are:

"trajectory of n" for the sequence

	t(k) =
		n if k = 0
		f(t(k-1)) if k > 0

"attractor" for a set S with f(S) = S. (that is, the set of values in a 
trajectory loop).

"fixpoint" for a single-element attractor (that is, {n} or n with f(n) = n).

"stopping point" for an element satisfying an artificial "stopping 
condition", such as n = 1 for Collatz or f(n+1) <= n for the sequence in 

So "stopping point with condition f(n+1) <= n" would avoid having to 
coin new terms.

franktaw at netscape.net wrote:
> Please don't call this the "destiny".  Destiny implies where you wind 
> up.  "First decrease" would be fine, if nobody comes up with something 
> better.
> Franklin T. Adams-Watters
> -----Original Message-----
> From: Maximilian Hasler <maximilian.hasler at gmail.com>
> I suggest a different notion of "destiny"
> (or someone may find a better name),
> namely, the first term in the trajectory under
> s(n)=sigma(n)-n
> which is not larger than its predecessor:
> ...
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