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

[Vladimir Shevelev]

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,
Vladimir

________________________________________
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.
> 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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