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

Max Alekseyev maxale at gmail.com
Sat Jul 11 13:55:47 CEST 2015

```Vladimir,
particular, (*) contains only a(1),...,a(n) but no any a(k) with k>n, so I
do not see how infiniteness of {a(n)} may help here.
Also, convergence (if it indeed takes place, which I doubt) needs to be
proved, and this is not so easy since the function A() (i.e. number of
divisors) has a hectic behavior.
Max
On Jul 10, 2015 12:15 PM, "Vladimir Shevelev" <shevelev at bgu.ac.il> wrote:

> I think that the sequence is indeed is an unique,
> according to the following simple argument. Let A=A000005.
> By the definition,
> a(n)=A(Sum_{i=1..n}a(i))=A(Sum_{i=1..n-1}a(i)+A(Sum_{i=1..n}a(i)))
> =A(Sum_{i=1..n-1}a(i)+A(Sum_{i=1..n-1}a(i)+A(Sum_{i=1..n}a(i))))=
> A(Sum_{i=1..n-1)}a(i)+A(Sum_{i=1..n-1)}a(i)+A(Sum_{i=1..n-1}a(i)+... (*)
> Since, an infinite sequence {a(n)} exists, by (*) a(n) is a function of
> a(1),...,a(n-1)
> and (*) should converge.
>
> Best regards,
>
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Max Alekseyev [
> maxale at gmail.com]
> Sent: 09 July 2015 23:03
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: apparently unique infinite sequences related to the
> sum of divisors
>
> 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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```