[seqfan] Re: Iterating some number-theoretic functions
Sean A. Irvine
sairvin at gmail.com
Mon Sep 18 23:20:22 CEST 2017
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
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
> 440 484 820592372224724578936331208771944114408888086190107072338829
> 496 413 584312582216816725920147810109410523690065654337049514240252
> 702 404 126251452114427769757166118472128627038379408031427558360064
> 737 402 283797546155754217301048520274749519198606895782115901466430
> 813 408 113022102836300508719080691707204575063141106900327215081396
> 828 403 455532144721661509141700260843225180989580666045895972886381
> 897 406 108940233399970132387522851290707383338719377108227439287474
> 905 404 100620027787656599681514901460240738592719142739691263136185
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