[seqfan] Re: apparently unique infinite sequences related to the sum of divisors

[Max Alekseyev]

In case anybody is interested, I've created a text file with all terms
of A259934 below 10^10 (with one term per line). There are 397,420,670
such terms and the file size is about 4.3GB. The compressed file is
available for download in either of two formats:

lrzip-archive of size 326MB:
https://drive.google.com/file/d/0B4NG9nYTfUlVd2pxZ0Y5U3h4U2s/view?usp=sharing

7zip-archive of size 450MB:
https://drive.google.com/file/d/0B4NG9nYTfUlVd2pPTEQwVGVvMEU/view?usp=sharing

Regards,
Max


On Thu, Jul 9, 2015 at 4:16 PM, Max Alekseyev <maxale at gmail.com> wrote:
> Neil,
>
> As of "finite amount of checking" -- imagine that we are at a branching
> point and we know that at least one of the branches is infinite. Then there
> are two cases:
>
> 1) Exactly one branch is infinite. We can easily detect which branch is
> infinite by showing that all but one branches are finite. The one which
> remains must be infinite. This check can be done in finite time.
>
> 2) Two or more branches are infinite. Here we may be in trouble to detect
> infinite branches.
>
> Luckily, type-2 branching points do not appear for quite a while (if ever)
> -- at least below 10^10 all branching points are of type 1.
>
> Regards,
> Max
>
>
> On Thu, Jul 9, 2015 at 4:03 PM, Max Alekseyev <maxale at gmail.com> wrote:
>>
>> Neil,
>>
>> No, it is not possible to reach a deadend for this sequence. Falcao proved
>> that if one starts with 0, then under an appropriate choice of next terms
>> the sequence will be infinite.
>> I've shown numerically that for the terms below 10^10 this choice is
>> unique (i.e., at each branching point only one of the branches is infinite).
>> It is possible that some time later there will be branching point giving
>> raise for two or more infinite branches (and then the sequence will not be
>> unique), but existence of such branching point is open.
>>
>> Falcao's proof is based on König's lemma -
>> http://en.wikipedia.org/wiki/K%C3%B6nig%27s_lemma
>>
>> Regards,
>> Max
>>
>>
>>
>>
>> On Thu, Jul 9, 2015 at 3:49 PM, Neil Sloane <njasloane at gmail.com> wrote:
>>>
>>> Max, You said:
>>>
>>> I have verified that all its values below 10^10 are uniquely
>>> defined (all possible bifurcations die sooner or later). So we can be
>>> sure
>>> that there is no dead end for A259934.
>>>
>>> Me: But it is possible that A259934 reaches a dead end after 10^100
>>> steps, right?  So then we would need to backtrack and change A259934.
>>>
>>> I don't see that any finite amount of checking will show we are on the
>>> right track!
>>>
>>> But I don't read Russian ....
>>>
>>> I agree this is a very nice sequence!
>>>
>>> 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 Thu, Jul 9, 2015 at 3:22 PM, Max Alekseyev <maxale at gmail.com> wrote:
>>>
>>> > Falcao proved that there is an infinite sequence starting with 2 (or 0
>>> > if
>>> > you like). I have verified that all its values below 10^10 are uniquely
>>> > defined (all possible bifurcations die sooner or later). So we can be
>>> > sure
>>> > that there is no deadend for A259934. It is an open question though if
>>> > there exists a viable bifurcation point.
>>> > Max
>>> >
>>> >
>>> > On Thu, Jul 9, 2015 at 3:06 PM, <israel at math.ubc.ca> wrote:
>>> >
>>> > > How do you know your values for A259934 are correct? It's true that
>>> > > a(n)
>>> > -
>>> > > d(a(n)) = a(n-1) for the listed values, but how do you know you don't
>>> > > run
>>> > > into a dead end? Falcao apparently (I don't read Russian) proved
>>> > > there is
>>> > > an infinite sequence, but there are also long finite sequences that
>>> > > have
>>> > > dead ends.
>>> > >
>>> > > Cheers,
>>> > > Robert
>>> > >
>>> > >
>>> > > On Jul 9 2015, Max Alekseyev wrote:
>>> > >
>>> > >  SeqFans,
>>> > >>
>>> > >> I'd like to draw your attention to two newly added nice sequences:
>>> > >>
>>> > >> http://oeis.org/A259934
>>> > >> http://oeis.org/A259935
>>> > >>
>>> > >> Comments are welcome.
>>> > >>
>>> > >> Regards,
>>> > >> Max
>>> > >>
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>>> > >>
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>>> > >>
>>> > >>
>>> > >>
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>>
>>
>