[seqfan] Re: 'Climb to a prime' in other bases

Mon Jun 19 03:44:44 CEST 2017

The smallest n for which the value of A230626 is unknown is now 12558,
a(12558) > 186 (by which time the composite is 99 decimal digits).  I will
be able to take this further, but progress is slowing down now that the
factorizations are becoming harder.

Prior to 12558 the hardest case is a(6356) = a(8621) = a(10442) = 145.

All the harder factors have been added to factordb.com

Sean.

On 18 June 2017 at 19:55, David Seal <david.j.seal at gwynmop.com> wrote:

> Following on from Sean A. Irvine's resolution of the trajectory of 234 and
> Neil Sloane's earlier suggestion of the sequence of numbers that don't
> reach a prime, I have now submitted A288847.
>
> Best regards,
>
> David
>
> > On 18 June 2017 at 03:30 "Sean A. Irvine" <sairvin at gmail.com> wrote:
> >
> >
> > I'm seeing a prime at step 104.  I'll make a b-file.
> >
> > On 18 June 2017 at 13:26, Neil Sloane <njasloane at gmail.com> wrote:
> >
> > > I created a new sequence A287878 for the trajectory of 234 under the
> map x
> > > -> A230625(x) (234 is the smallest number whose destiny is presently
> > > unknown). It needs a b-file.
> > >
> > > For the analogous map x-> A048985(x), the entry A048986 says that all
> > > numbers <= 2294 end at 1 or a prime,
> > > but the destiny of 2295 is unknown. The trajectory of 2295 should
> probably
> > > also be entered as a new sequence, with a b-file, in case anyone would
> like
> > > to help.
> > >
> > >
> > > 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 Fri, Jun 16, 2017 at 11:43 PM, David Seal <david.j.seal at gwynmop.com
> >
> > > wrote:
> > >
> > > [... first part of post omitted for brevity ...]
> > >
> > > > > It would be interesting to see the list of numbers that don't
> reach 1
> > > or
> > > > a
> > > > > prime.  That is, the numbers that are fixed (and composite), or go
> to a
> > > > > composite fixed point, or go into a loop.  If you have this list,
> could
> > > > you
> > > > > submit it as a new sequence?
> > > >
> > > > My program didn't give me that information, but I've produced a
> modified
> > > > version that does. Unfortunately, though, it's only given me one
> term of
> > > > the new sequence...
> > > >
> > > > That term is 217, which enters the 1007 <-> 1269 loop after 3 steps:
> > > >
> > > > 217 = 7 * 31 = binary 111 * 11111 -> binary 11111111 = 255
> > > > 255 = 3 * 5 * 17 = binary 11 * 101 * 10001 -> binary 1110110001 = 945
> > > > 945 = 3^3 * 5 * 7 = binary 11^11 * 101 * 111 -> binary 1111101111 =
> 1007
> > > >
> > > > All other values up to 233 get to primes, but I have not yet been
> able to
> > > > determine what happens to 234, beyond that it grows to >= 2^63.
> Whether
> > > it
> > > > ends up at a prime, in one of the known loops, in some as-yet-unknown
> > > loop,
> > > > or just growing forever, I don't know, and so I don't know whether
> it's
> > > in
> > > > your proposed sequence or not.
> > > >
> > > > Values like 234 are not uncommon. I've looked at the numbers 1-10000,
> > > with
> > > > the following results:
> > > >
> > > > 9808 numbers were 1 or got to primes
> > > > 42 numbers enter loops (other than primes going to themselves)
> > > > 150 numbers went to values >= 2^63 without being resolved
> > > >
> > > > What I can tell you is that your proposed sequence starts:
> > > >
> > > > 217, 234?, 255, 408?, 420?, 446, 545?, 558, 564?, 595?, 668?, 678?,
> 717,
> > > > 735, 749?, 775, 777?, 830?, 844?, 861?, 882?, 904?, 918?, 945, 950?,
> > > 956?,
> > > > 958, 1003?, 1007, ...
> > > >
> > > > where question marks indicate terms that might or might not actually
> be
> > > in
> > > > the sequence.
> > > >
> > > > Best regards,
> > > >
> > > > David
> > > >
> > > >
> > > > > On 15 June 2017 at 20:35 Neil Sloane <njasloane at gmail.com> wrote:
> > > > >
> > > > >
> > > > > David Seal:  Nice work!
> > > > >
> > > > > It turns out that the base-2 version is already in the OEIS,
> > > > > although without your counterexample, which I have now added.
> > > > >
> > > > > I also added some sequences that were missing, so the base-2
> problem is
> > > > now
> > > > > described in the following 5 entries:
> > > > > A230625 & A287874 for the basic map,
> > > > > and, for what you reach when the map is
> > > > > iterated, A230626, A230627, A287875.
> > > > >
> > > > > There is a slight awkwardness in A287875, because the escape
> clause,
> > > > which
> > > > > says we write -1 if we don't reach 1 or a prime, is tricky to deal
> with
> > > > in
> > > > > base 2.  So the escape clause is "a(n) = -1 in decimal if ...", and
> > > > > otherwise "a(n) = the prime reached, or 1, written in binary".
> > > > >
> > > > > Did you check to see if 255987 is the smallest number that is
> fixed in
> > > > base
> > > > > 2?  I assume so, but it wasn't clear from your message.
> > > > >
> > > > > It would be interesting to see the list of numbers that don't
> reach 1
> > > or
> > > > a
> > > > > prime.  That is, the numbers that are fixed (and composite), or go
> to a
> > > > > composite fixed point, or go into a loop.  If you have this list,
> could
> > > > you
> > > > > submit it as a new sequence?
> > > > >
> > > > >
> > > > > 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
> > > > >
> > > > > [... earlier post omitted for brevity ...]
> > > > >
> > > > > --
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