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

Bob Selcoe rselcoe at entouchonline.net
Sat Jul 29 08:18:53 CEST 2017

```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/
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```