[seqfan] Re: Iterating some number-theoretic functions

Hugo Pfoertner yae9911 at gmail.com
Sun Sep 3 19:02:10 CEST 2017

I have run some computations for problem (c). Results can be seen in
https://oeis.org/A289997 and https://oeis.org/A290001

The iteration will not start for those k, where already the first sum is
odd, and that is for k=n^2 or k=2*n^2

There are some long trajectories, for which I stopped the search when the
sum exceeded
my table of 10^6 terms of sigma(n) and phi(n). Examples are the start values
270, 290, 308,  326, 327, 328, 352, 369, 390, 393-396, ..., 570, 572, and
many more

Hugo Pfoertner

On Sun, Sep 3, 2017 at 6:49 AM, Neil Sloane <njasloane at gmail.com> wrote:

> Dear Sequence Fans,
> Let sigma = A203, phi = A10, psi = A1615. Richard Guy's Unsolved Problems
> in Number Theory, 3rd ed., (UPNT) Section B41, pp. 147-150, mentions
> several problems related to iterating these functions that caught my eye
> recently.
> ...
> (c) If we iterate k -> (sigma(k)+phi(k))/2 (same ref.), sometimes we reach
> a fraction, when we say the trajectory has fractured, and we quit.
> Question: what are the starting values n whos trajectory doesn't fracture?
> Also, Richard asks if there are starting values which increase indefinitely
> without fracturing.

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