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

[Max Alekseyev]

Neil,

The question on whether every odd number is present in A109732 can be
reformulated as follows.
Can any odd number m be transformed into 1 with maps: m -> (m-1)/2 (only if
the result is integer) and m -> 3m applied in some order?
It is clear that even numbers cannot appear in such a transformation, since
they would remain even forever and thus not reach 1.

Replacing m with n = (m+1)/2, we get an equivalent question:
Can any number n be transformed into 1 with maps: n -> n/2 (only if n is
even) and n -> 3n-1 applied in some order?

It is worth to mention that the affirmative answer to this question would
follow from the 3x-1 variation of Collatz conjecture. Namely, it states
that the map x -> x/2 (for even x) and x -> 3x-1 (for odd x) eventually
reaches one of the three cycles: (1,2), (5, ...) http://oeis.org/A003079 of
length 5, or (17,...) http://oeis.org/A003124 of length 17.
In contrast to this 3x-1 variation, we have a freedom of choosing a map
(out of the two) to apply (the only restriction is that n -> n/2 can be
applied only if n is even). With this freedom, we can transform 5 and 17
from the non-trivial cycles of the 3x-1 problem to 1:

(5, *14*, 7, 20, 10, 29, 86, 43, 128, 64, 32, 16, 8, 4, 2, 1)

(17, 50, 25, 74, 37, 110, 55, 164, 82, 41, 122, 61, 182, 91, 272, 136, 68,
203, 608, 304, 152, 76, 38, 19, 56, 28, *14*, ... as before)

That is, under the 3x-1 variation of Collatz conjecture, we can transform
any number either to 1, 5, or 17. In the latter two cases, we can proceed
further as explained above and still reach 1.

Regards,
Max













On Thu, Aug 27, 2015 at 1:39 PM, Neil Sloane <njasloane at gmail.com> wrote:

> 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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