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

Mon Aug 31 19:05:58 CEST 2015

Thank you, Max, I indeed missed that.
On the other hand, I submitted a proof
by induction of the conjectural explicit
formula for a(n) in A261728.

Best,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Max Alekseyev [maxale at gmail.com]
Sent: 30 August 2015 18:21
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: An old sequence from van der Poorten

Vladimir,

"2) If N+1 = 6*t+1, then 2*t<n and, by IS,

2*t=>N+1=>...=>1;"

The first step here violates the (3n+1) rule. It has to go as "2*t => t =>
...".
So your proof is flawed.

Regards,
Max

On Sun, Aug 30, 2015 at 10:46 AM, Vladimir Shevelev <shevelev at bgu.ac.il>
wrote:

> Dear Seq Fans,
>
> I submitted A261728:
>
> a(1)=1; a(2*n) = 3*n; for odd n>1, a(n) is the smallest number not already
> present which is entailed by the rules (i) k present => 3*k+1 present; (ii)
> 2*k present => k present.
>
>  DATA  1, 3, 4, 6, 2, 9, 7, 12, 10, 15, 5, 18, 13, 21, 16, 24, 8, 27, 19,
> 30, 22, 33, 11, 36, 25, 39, 28, 42, 14, 45, 31, 48, 34, 51, 17, 54, 37, 57,
> 40, 60, 20, 63, 43, 66, 46, 69, 23, 72, 49, 75, 52, 78, 26, 81, 55, 84, 58,
> 87, 29, 90, 61, 93, 64, 96, 32, 99, 67, 102, 70, 105, 35, 108, 73, 111, 76,
> 114, 38, 117, 79
>
> Conjecture. The only fixed points are 6*k+1, k>=0; if n==3 (mod 6), then
> a(n) = n+1; if n==5 (mod 6), then a(n) = (n-1)/2.
>
> If the conjecture is true, then the sequence
> is a permutation of the positive integers.
>
> On the other hand, the statement 'the sequence is a permutation of the
> positive integers' is equivalent to the (3*n+1)-conjecture.
>
> Proof. Indeed, if (3*n+1)-conjecture is true
>
> then, according to the rule, the sequence is a permutaion of the positive
> numbers. Conversely, we use induction. Let for all m<=N, the
> (3*n+1)-conjecture is true. Consider N+1.
>
> 1) If N+1 is even = 2^k*t, where t is odd.
>
> Since t<N, then, beginning with N+1=>t etc., by the rule of
> (3*n+1)-problem, by the inductional supposition (IS), we reach 1.
>
> 2) If N+1 = 6*t+1, then 2*t<n and, by IS,
>
> 2*t=>N+1=>...=>1;
>
> 3)Finally, let N+1 be odd of the form 6*t+5. Then 4*t+3<N. Therefore, by
> IS, we have 4*t+3=>12*t+10=>N+1=>...=>1. QED
>
> Thus the above Conjecture  implies the (3*n+1)-conjecture.
>
> Best regards,
> Vladimir
>
> ________________________________________
> From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir
> Shevelev [shevelev at exchange.bgu.ac.il]
> Sent: 29 August 2015 20:45
> To: Sequence Fanatics Discussion list
> Subject: [seqfan] Re: An old sequence from van der Poorten
>
> 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
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
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>
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>
> Seqfan Mailing list - http://list.seqfan.eu/
>

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