[seqfan] Re: A098550.

Fri Dec 5 00:59:12 CET 2014

Hi All,

I could not work around one criticism of Neil Sloane. My proposed proof
does not show the impossibility of infinite D-loops.

It is more difficult to show the necessity of ascending primes, because
when

A[n-2] = p[i] p[m]

with p[m] the smallest missing prime, it could be the case that

A[n] = x < p[m]

where p[i] is a factor of x. This does happen for low n, but apparently
does not happen for larger n, if my conjecture about http://oeis.org/A251416
holds. There is decent evidence that this conjecture is true, which can be
visualized by plotting two sequences:

http://oeis.org/A251239

And another new sequence, defined as first m such that the range of numbers
p(n) to p(n+1) is completely included in the sequence A098550(m).

The first terms are:

0, 4, 10, 16, 12, 14, 31, 20, 33, 39, 37, 48, 44, 56, 54, 69, 71, 73,
80, 75, 95, 89, 97, 104, 110, 115, 112, 117, 128, 143, 139, 147, 149,
161, 176, 178, 180, 182, 187, 192, 205, 207, 209, 211, 216, 229, 241,
245, 248...

Around n = 0 this sequence is somewhat mixed up with A251239, but as n
increases the ratio of the approximately linear slope is about 2, and the
difference between series increases.

Variation in the slope is not much. Taking the variation around a linear
regression up to value a(n) as a predictor of value a(n+1), it apparently
becomes less probable for the sequence to reach prime p(m) before filling
the range between p(m) and p(m+1). For this to happen the new range filling
sequence would have to spike sharply upward.

It seems unlikely, but it's difficult to say with primes.

If any one of the following can be shown:

1. another prime always occurs in the sequence
2. infinite D-loops do not occur
3. the sequence never enters state D

Then it will at least be possible to write another draft of the proof, and
I think that draft would turn out true.

Thanks,

Brad

On Wed, Dec 3, 2014 at 4:03 PM, Neil Sloane <njasloane at gmail.com> wrote:

> 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.
> > >
> > > _______________________________________________
> > >
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> >
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> >
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> >
>
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