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

Max Alekseyev maxale at gmail.com
Fri Sep 4 19:05:27 CEST 2015

```Just a follow-up.

I've posted a question on such a relaxed variation of Collatz to MO:
http://mathoverflow.net/questions/216358/relaxed-collatz-3x1-conjecture
and discovered that the length of shortest transformation for n is present
in the OEIS as http://oeis.org/A127885
I've added a similar sequence for the relaxed 3x-1 version as
http://oeis.org/A261870

Regards,
Max

On Fri, Aug 28, 2015 at 2:43 PM, Max Alekseyev <maxale at gmail.com> wrote:

> 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
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> 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.