[seqfan] Re: An old sequence from van der Poorten

[Max Alekseyev]

Vladimir,

You said "A261690 is a permutation of numbers not divisible by 3 =>
(n-1)/3-conjecture", but this is not obvious either.
A261690 assumes application of the two maps in arbitrary order. In your
(n-1)/3-conjecture (as well as in the 3n+1 conjecture) the order is fixed.

Max


On Sat, Aug 29, 2015 at 1:45 PM, Vladimir Shevelev <shevelev at bgu.ac.il>
wrote:

> Dear Max,
>
> You are right, indeed there are difficulties to prove that. So, in A261690
> I did the following comment.
>
> "29 Aug 2015 Max Alekseyev noted that, while
>
> (3*n+1)-cojecture indeed implies that the sequence is a permutation of the
> positive integers not divisible by 3, but the opposite statement is far
> from obvious
>
> at all. The author cannot yet prove this, so his previous comment is only
> a conjecture. In connection with this,
>
> consider the following conjecture which naturally could be called
> (n-1)/3-conjecture.
> Let n be any number not divisible by 3. If n==1 (mod 3) and (n-1)/3 is not
> divisible by 3, then set n_1 = (n-1)/3. Otherwise set n_1 = 2*n.
>
> Conjecture. There exists an iteration n_m = 1.
> Example: 19->38->76->25->8->16->5->10->20->40->13->
> 4->1.
>
> Then we have: (3*n+1)-conjecture =>
> A261690 is a permutation of numbers not divisible by 3
> => (n-1)/3-conjecture.
> Does (n-1)/3-conjecture imply (3*n+1)-conjecture? "
>
> Best,
> Vladimir
>
> ________________________________________
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Max Alekseyev [
> maxale at gmail.com]
> Sent: 29 August 2015 04:40
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: An old sequence from van der Poorten
>
> Vladimir,
> I do not see equivalence here. The 3x+1 conjecture indeed implies that
> A261690 is a permutation, but in the opposite direction it is not obvious
> at all. Can you prove it?
> Max
> On Aug 28, 2015 2:50 PM, "Vladimir Shevelev" <shevelev at bgu.ac.il> wrote:
>
> > Dear Neil,
> >
> >  I submitted A261690 which is an
> > analog of A109732 such that the
> > statement ' the sequence is a
> > permutation of the positive integers
> > not divisible by 3' is equivalent to
> > the (3*n+1)-conjecture on numbers
> > not divisible by 3.
> > So I think that Van der Poorten's
> > question is in the same degree unprovable
> > as the (3*n+1)-conjecture.
> >
> >
> > Best regards,
> > Vladimir
> >
> > ________________________________________
> > From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Neil Sloane [
> > njasloane at gmail.com]
> > Sent: 27 August 2015 20:39
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] An old sequence from van der Poorten
> >
> > Dear Seq Fans:
> >
> > An old sequence suggested by a posting by Alf van der Poorten is A109732:
> > a(1) = 1; for n>1, a(n) is the smallest number not already present which
> is
> > entailed by the rules (i) k present => 2k+1 present; (ii) 3k present => k
> > present.
> > The open question is whether every odd number appears.
> >
> > It seems that numbers of the form 2^k+1 take an exceptionally long time
> to
> > appear - see A261414, which needs more terms. In particular, when does
> 1025
> > appear in A109732?
> >
> >
> > 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
> >
> > _______________________________________________
> >
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> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
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