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

Neil Sloane njasloane at gmail.com
Fri Jul 10 00:30:19 CEST 2015

```Max, Thanks for the explanation - that is
a convincing argument!

Some time, could you add b-files to those two sequences?

Also maybe add the program you used, possibly as a .txt file
(it doesn't need to be a beautiful program, but it
is always good to see how the sequence was generated, so that others can
check 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.
Email: njasloane at gmail.com

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.
> >> 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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> >
> >
>
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```