[seqfan] Re: A098550.

[Benoît Jubin]

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.
>
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