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From: howard@metheus.UUCP (Howard A. Landman)
Newsgroups: net.math
Subject: Re: Iterated sums of digits of divisors
Message-ID: <254@metheus.UUCP>
Date: Fri, 27-Jul-84 19:45:48 EDT
Article-I.D.: metheus.254
Posted: Fri Jul 27 19:45:48 1984
Date-Received: Mon, 30-Jul-84 00:01:31 EDT
References: <2976@ecsvax.UUCP>
Organization: Metheus, Portland Oregon
Lines: 18

Here is an outline of a proof which I am too lazy right now to complete.

(1) Prove that for some N, for all M > N, K(M) < M.
    This establishes that the sequences for all sufficiently large
    numbers will eventually enter (and STAY in) the set of numbers < N.
    Thus we only need to consider sequences which start with < N.

(2) Any sequence which remains in a finite set must eventually repeat an
    element.  Because of the way this series is calculated, such a repeat
    means that the series will then repeat a sequence or a single number
    forever.

(3) Prove by exhaustive examination of the cases that there are no such
    cycles other than 15.  (If the statement is false, of course, it is
    here that you will discover the counterexample.)

	Howard A. Landman
	ogcvax!metheus!howard