[seqfan] Re: Iterating some number-theoretic functions

[Sean A. Irvine]

Andrew Booker makes the following observations:

I would guess that asymptotically 100% of positive integers have unbounded
trajectory, and it might be possible to prove something along those lines.

Note that (sigma(n)+phi(n))/2 is an integer unless n is a square or twice a
square, and those are very rare among large numbers. Further, if n>1 and
(sigma(n)+phi(n))/2 is odd then n must be of the form pm^2 or 2pm^2 for an
odd prime p. Combining this with some sieve theory, one can show that the
number of composite n<=x with (sigma(n)+phi(n))/2 prime is O(x/log^2x).
Since the map tends to increase geometrically and sum 1/k^2 converges, this
suggests that a typical large composite has little chance of ever reaching
a prime.

Sean.


On 14 September 2017 at 16:42, Sean A. Irvine <sairvin at gmail.com> wrote:

> 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 766431583175462762130381515662187930626060289448722569860560
> 0248337350669671380953658464325279694429699208993393252810106664744901740672517008
>
> 440 484 820592372224724578936331208771944114408888086190107072338829
> 838271921194090028163759134652444477758963320036027410363866196889253338112
>
> 496 413 584312582216816725920147810109410523690065654337049514240252
> 7907753582982754877434126465705917383073237325824
> 702 404 126251452114427769757166118472128627038379408031427558360064
> 354641637918168792707424465732797790457902669435072
> 737 402 283797546155754217301048520274749519198606895782115901466430
> 74098002112474486185406438722917960742713527040
> 813 408 113022102836300508719080691707204575063141106900327215081396
> 1006171199412698705455907629229684455325220217689600
> 828 403 455532144721661509141700260843225180989580666045895972886381
> 942233201091585638600868459337792419151007988203520
> 897 406 108940233399970132387522851290707383338719377108227439287474
> 8695473478395781640270557738150084391944706967040
> 905 404 100620027787656599681514901460240738592719142739691263136185
> 0971140981478254530295756493557282131306214400
>
> Sean.
>
>