[seqfan] Re: Iterating some number-theoretic functions

[Sean A. Irvine]

A summary of computation done for the (sigma(n)+phi(n))/2 iterated map.

For n <= 1000, there are 9 main trajectories for which the outcome is
unknown (i.e. repeated iteration of the map seems to always give an
integer).  There are actually more than 9 starting points, but a bunch of
them converge and we end up with 9 main lines. There is a picture in
A291790 which shows this.

The smallest members of each of these nine trajectories are 270, 440, 496,
702, 737, 813, 828, 897, 905.

I have iterated all of the 9 trajectories for at least 400 steps as shown
below.  I had been hoping that at least one of these would have fractured
by now, but it was not to be.  I think it is over to the theory guys now to
explain :-)

start value, number of known terms, last known term
270 515
7664315831754627621303815156621879306260602894487225698605600248337350669671380953658464325279694429699208993393252810106664744901740672517008

440 484
820592372224724578936331208771944114408888086190107072338829838271921194090028163759134652444477758963320036027410363866196889253338112

496 413
5843125822168167259201478101094105236900656543370495142402527907753582982754877434126465705917383073237325824

702 404
126251452114427769757166118472128627038379408031427558360064354641637918168792707424465732797790457902669435072

737 402
28379754615575421730104852027474951919860689578211590146643074098002112474486185406438722917960742713527040

813 408
1130221028363005087190806917072045750631411069003272150813961006171199412698705455907629229684455325220217689600

828 403
455532144721661509141700260843225180989580666045895972886381942233201091585638600868459337792419151007988203520

897 406
1089402333999701323875228512907073833387193771082274392874748695473478395781640270557738150084391944706967040

905 404
1006200277876565996815149014602407385927191427396912631361850971140981478254530295756493557282131306214400


Sean.