[seqfan] Re: Iterating "smallest odd prime divisor of n^2 + 1"

[Neil Sloane]

Perhaps there are should be two new sequences (at least):

- number of steps to reach either 5 or 13 when starting with n
- largest number in the trajectory of n

(with a cross-reference to A125256 to make them easy to find)

Could someone create them?


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


On Sat, Nov 4, 2017 at 9:54 AM, Robert Gerbicz <robert.gerbicz at gmail.com>
wrote:

> Not that nice, all of these converged pretty easily, for example:
> ? g(4949)
> 0 4949
> 1 12246301
> 2 74985944091301
> 3 2811445905631859677311936301
> 4 3952114040147073815012888046973669539943086904940781301
> 5 101
> 6 5101
> 7 13
> 8 5
>
>
> 2017-11-04 13:42 GMT+01:00 Neil Sloane <njasloane at gmail.com>:
>
> > > for n=4949, 6051, 9751 and 12281, at least, the map "n -> smallest odd
> > prime divisor of n^2+1" diverges.
> >
> > Very nice!  Could you explain the proof?
> >
> > 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
> >
> >
> > On Sat, Nov 4, 2017 at 7:08 AM, Luca Petrone via SeqFan <
> > seqfan at list.seqfan.eu> wrote:
> >
> > > Dear All,
> > >
> > > for n=4949, 6051, 9751 and 12281, at least, the map "n -> smallest odd
> > > prime divisor of n^2+1" diverges.
> > >
> > > Regards,
> > >
> > > Luca Petrone
> > >
> > > >
> > > >     Il 4 novembre 2017 alle 11.45 Frank Adams-Watters via SeqFan <
> > > seqfan at list.seqfan.eu> ha scritto:
> > > >
> > > >     It was definitely my impression that it did always fall into that
> > > loop when I looked at the question in conjunction with my edit to the
> > > related sequence A031439 back in 06. I didn't have any idea of how to
> > prove
> > > it, however.
> > > >
> > > >     Probabilistically, it seems likely: you only have to get a term
> > > congruent to 2 or 3 mod 5, and you immediately fall into the loop.
> > > >
> > > >     Franklin T. Adams-Watters
> > > >
> > > >     -----Original Message-----
> > > >     From: Neil Sloane <njasloane at gmail.com>
> > > >     To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > > >     Sent: Fri, Nov 3, 2017 11:59 pm
> > > >     Subject: [seqfan] Iterating "smallest odd prime divisor of n^2 +
> 1"
> > > >
> > > >     Dear Seq Fans, While I was at Hofstra Univ. the other day, Zoran
> > > Sunik
> > > >     asked, if you iterate the map "n -> smallest odd prime divisor of
> > > n^2+1",
> > > >     do you always end in the 2-cycle (5 <-> 13) ?Does anyone know?
> > > >     See A125256 for the map, and also its bisections A256970and
> > A293958.
> > > >     If this is true, then there could be a sequence giving the number
> > of
> > > >     iterations needed to reach the loop, or to reach any loop if
> there
> > > are
> > > >     others ...
> > > >
> > > >     --Seqfan Mailing list - http://list.seqfan.eu/
> > > >
> > > >     --
> > > >     Seqfan Mailing list - http://list.seqfan.eu/
> > > >
> > >
> > > --
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> > >
> >
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