[seqfan] Re: A098550.

[Neil Sloane]

Concerning Brad Klee's posting from 15 hours ago, there are several steps
in his argument that I could not follow.

One point concerns the sentence:

"Recycling the logic used above let the smallest un-included number a[n] = x,
and count p[??] to equal N[ p[??] ] all the prime numbers co-prime to x
satisfying
p[??] < x < p[m].
where p[m] is the smallest prime number larger than x."

Is there a word missing?

Another point concerns the sentence
"But this conjecture does not matter because prime numbers occur throughout the
sequence and primes are co-prime to all numbers, so an infinite D-loop cannot
exist."

Here what worries me. We know that every prime p *divides* at least one
term in A098550 (see the comments in that entry for proof), but we don't
yet know that every prime appears naked.

So "prime numbers occur throughout the sequence and primes are co-prime to
all numbers" doesn't seem to make sense.

So, continuing this objection, why cannot we have an infinite sequence of
D-states where multiples of 2 alternate with multiples of 3?


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 Wed, Dec 3, 2014 at 12:10 PM, Benoît Jubin <benoit.jubin at gmail.com>
wrote:

> I do not really see the point in studying these latter "dynamical"
> sequences. Studying the dynamics (fixed points, orbits...) is
> interesting for a function from a space to itself, but here this is
> not the case: A098550 is from \N the first infinite ordinal to the
> ordered multiplicative monoid \N-{0}. I agree that it is sometimes
> useful to mix the structures and discover a new phenomenon, but I
> think here it is not the case. For instance, defining b by
> b(n)=A098550(n+1), we obtain morally the same sequence, but the fixed
> points, iterates and lengths of loops will be very different. On the
> other hand, the adherence values of b(n)/n (finite or infinite) will
> be the same, so this initial question seems more interesting to study
> (another hint for the interest of that question is that it was asked
> by FTAW).
>
> Rather, we could numerically test if A098550(n)/n has finite adherence
> values and what they are. I suggest to study the apparently simpler
> behavior of A098548(n)/n: does it have a finite limit and what is it?
> Can we exclude a behavior like c.n.ln(ln(n))? A naive approach proves
> the very weak inequalities
> 3n < A098548(n) < 1.001^(sqrt(3/2)^n)
> (by a case study, one can probably replace 3 by 3.75 and with more
> effort higher, and obviously the value 1.001 is not important). Can
> one prove for instance that
> A098548(n) = O(n^2) or O(n.ln(n))?
>
> I do not mean to understimate any kind of work, and my first paragraph
> might be due to misunderstandings. I'm simply trying to focus the
> efforts towards what I think is more interesting.
>
> Thanks
>
> Benoît
>
>
> On Wed, Dec 3, 2014 at 8:57 AM, Hans Havermann <gladhobo at teksavvy.com>
> wrote:
> > Neil Sloane:
> >
> >> They are all mentioned in the Cross-references section of A098550.
> >
> > Good stuff. I see that you OEIS-internalized a link of my "portion of
> the trajectory containing 11" in http://oeis.org/A251412 which is fine,
> but I have just put up (an external)
> http://chesswanks.com/num/a098550loops&chains.txt that provides
> (currently 26 and 305, respectively) loops and unresolved chains for map n
> -> A098550(n) trajectories, including an already-extended trajectory for
> 11. By referring to that (instead), A251412 can take advantage of my
> updates as I grow my database over the coming days/weeks.
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>