# [seqfan] Re: corollary to the Collatz conjecture?

jean-paul allouche jean-paul.allouche at imj-prg.fr
Sat Jul 29 11:21:01 CEST 2017

Hi

I understand. Actually there is a generalized Collatz conjecture
due to Hasse and described in a paper by M\"oller, and one by Heppner
in 1978 (and one by myself the same year), where you call \varphi a map
that sends Z to a complete set of residues modulo d and you do x --> x/d
if d divides x, and x --> (nx - \varphi(nx)) otherwise. The conjecture
is that
for n < d^{d/(d-1)} every integer goes to a loop and the number of loops is
finite, and that for n > d^{d/(d-1)} there exist infinite trajectories.
I am not sure whether the case you are interested in was addressed
explicitly
since then. The ultimate source is Lagarias's book
http://bookstore.ams.org/mbk-78

best wishes
jean-paul

Le 29/07/17 à 08:18, Bob Selcoe a écrit :
> Hi Jean-Paul and Seqfans,
>
> Sorry if I weren't clear enough.  I'm taking an infinite number of
> different functions:  x --> x/2 if x is even and (3x+3^k) if x is odd
> for all fixed k >=0 .  So I conjecture that not only x --> x/2 if x is
> even and 3x+1 if x is odd eventually reaches terminal loop [1,4,2,1]
> for all starting values of x (i.e., the Collatz conjecture), but also
> that 3x+3 if x is odd reaches loop [3,12,6,3], 3x+9 reaches loop
> [9,36,18,9], 3x+27 reaches loop [27,108,54,27], etc., for all starting
> values of x.
>
> Hope that's clearer.
>
> Cheers,
> Bob
> PS Olivier - when I hit "Reply" the recipient was just Jean-Paul; when
> I hit "Reply All" it went to the general Seqfan list.  That's never
> happened before - not sure if it's a glitch or is that now intentional??
>
>
> --------------------------------------------------
> From: "jean-paul allouche" <jean-paul.allouche at imj-prg.fr>
> Sent: Saturday, July 29, 2017 12:20 AM
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: corollary to the Collatz conjecture?
>
>> Dear Bob
>>
>> I am not sure I understand your point. Do you tak the *same* map?
>> (i.e., x --> x/2 if x is even and (3x+1) if x is odd).
>> If so, the only (conjectural) loop is  (1,4,2,1).
>> If not, which map are you looking at?
>> best
>> jpa
>>
>> Le 28/07/17 à 17:47, Bob Selcoe a écrit :
>>> Hi Seqfans,
>>>
>>> I'm wondering if there is a corollary to the Collatz conjecture for
>>> k >= 0, that 3x + 3^k eventually terminate with a 4-term loop [3^k
>>> => 4*3^k => 2*3^k => 3^k].
>>>
>>> So 3x+1 is simply k=0 with terminal loop [1,4,2,1];  k=1 is 3x+3
>>> with loop [3,12,6,3];  k=2 is 3x+9 with loop [9,36,18,9]; etc.
>>>   There do not appear to be any OEIS entries pertaining to k > 0.
>>>
>>>   I don't know how to program so I can't systematically test the
>>> conjecture, but doing some sequences by hand using this online tool
>>> http://www.dcode.fr/collatz-conjecture suggests it may hold.
>>>
>>> Does anyone know if this has been tested, or if there are any papers
>>>
>>> Cheers,
>>> Bob Selcoe
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
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