[seqfan] Re: Iterating some number-theoretic functions

Neil Sloane njasloane at gmail.com
Mon Sep 4 02:10:31 CEST 2017


Here is my previous message, corrected:

Concerning iterating the map k->(phi(n)+sigma(n))/2, starting at n.

Thanks to Hugo, we can now say a bit more. There are several possibilities:
- if n is a square > 1 or twice a square > 2, then we get a fraction after
one step and we die (A028982 \ {1,2})
- we get a fraction after more than one step and we die (A290001).
The union of these first two possibilities is A291791.  What ARE these
numbers?

- the trajectory never reaches a fraction (A289997). There are 3 subcases:
--- the trajectory reaches a prime (all primes are fixed points) - what are
these values of n?
--- the trajectory diverges (A291790, although proofs are lacking)
--- the trajectory goes into a nontrivial cycle (no examples are known)

The first trajectory that appears to be integral and unbounded is that of
270, see A291789. Perhaps if that were understood it would help
prove that all the terms of A291790 are correct (just as the trajectory of
45, A291787, for one of the other problems, has the recurrence a(n)=2*a(n-8)
and is therefore unbounded)

So when we start with n = 270, how does it know that it is never going to
produce a fraction?

Its trajectory (A291789) starts
270, 396, 606, 712, 851, 852, 1148, 1416, 2032, 2488, 2960, 4110, 5512,
6918,...
and the prime factorizations are interesting but not all that helpful.

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com


On Sun, Sep 3, 2017 at 7:48 PM, Neil Sloane <njasloane at gmail.com> wrote:

> There were errors in my last message.  Hang on while I redo it.
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098 <(732)%20828-6098>; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
>
> On Sun, Sep 3, 2017 at 3:28 PM, Neil Sloane <njasloane at gmail.com> wrote:
>
>> Concerning iterating the map k->(phi(n)+sigma(n))/2, starting at n.
>>
>> Thanks to Hugo, we can now say a bit more. There are several
>> possibilities:
>> - if n is a square or twice a square, then we get a fraction after one
>> step and we die (A028982)
>> - we get a fraction after more than one step and we die (A290001)
>> - the trajectory never reaches a fraction (A289997). There are 3 subcases:
>> --- the trajectory reaches a prime (all primes are fixed points) - what
>> are these values of n?
>> --- the trajectory diverges (A291790, although proofs are lacking)
>> --- the trajectory goes into a nontrivial cycle (no examples are known)
>>
>> The first trajectory that appears to be integral and unbounded is that of
>> 270, see A291789. Perhaps if that were understood it would help
>> prove that all the terms of A291790 are correct (just as the trajectory
>> of 45, A291787, for one of the other problems, has the recurrence
>> a(n)=2*a(n-8)
>> and is therefore unbounded)
>>
>>
>>
>> Best regards
>> Neil
>>
>> Neil J. A. Sloane, President, OEIS Foundation.
>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
>> Phone: 732 828 6098 <(732)%20828-6098>; home page: http://NeilSloane.com
>> Email: njasloane at gmail.com
>>
>>
>> On Sun, Sep 3, 2017 at 1:02 PM, Hugo Pfoertner <yae9911 at gmail.com> wrote:
>>
>>> 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
>>> https://oeis.org/A028982
>>>
>>> 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.
>>> >
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>>
>>
>>
>


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