[seqfan] Re: Near-linear sequence

Tue Sep 16 21:23:18 CEST 2014

Benoit, Very nice! I just approved the changes to A101402.

In the third line from the bottom, it should say:

It is proved in the entry for A246878 that ...

Can you change it?

Neil

On Tue, Sep 16, 2014 at 3:17 PM, Charles Greathouse <
charles.greathouse at case.edu> wrote:

> Thank you Benoît!
>
> Charles Greathouse
> Analyst/Programmer
> Case Western Reserve University
>
> On Tue, Sep 16, 2014 at 3:07 PM, Benoît Jubin <benoit.jubin at gmail.com>
> wrote:
>
> > Dear Charles, Neil, and seqfans,
> > Concerning the A101402 entry, I just made the modifications as Charles
> > proposed.
> > Best regards,
> > Benoit
> >
> > On Mon, Sep 8, 2014 at 9:08 PM, Charles Greathouse
> > <charles.greathouse at case.edu> wrote:
> > > Good job! I hope you add this to the sequence -- probably as an
> auxiliary
> > > file (as you suggest) with a formula referencing it ("a(n) = Theta(n),
> > see
> > > Jubin link. - ~~~~").
> > >
> > > The next step, if anyone's up for it, would be to prove a(n) ~ n which
> > > would require tightening both proofs.
> > >
> > > Charles Greathouse
> > > Analyst/Programmer
> > > Case Western Reserve University
> > >
> > >
> > > On Mon, Sep 8, 2014 at 1:26 PM, Benoît Jubin <benoit.jubin at gmail.com>
> > wrote:
> > >
> > >> I proved a weak form of Charles Greathouse's conjecture, namely,
> > >> A101402(n) = Theta(n). First, one has A101402(n) = Theta(A246878(n))
> > >> by the argument I roughly gave above (which I can rewrite in the entry
> > >> for the sequence, or given the length, in an auxiliary file?). Second,
> > >> I proved in the entry of A246878 that A246878(n) = Theta(n), with
> > >> explicit constants. By looking carefully at the first argument (in
> > >> particular looking quantitatively at the approximation of the Lambert
> > >> function), it might be possible to give explicit constants in
> > >> A101402(n) = Theta(n).
> > >>
> > >> Benoît
> > >>
> > >>
> > >> On Mon, Sep 1, 2014 at 8:30 AM, Aai <agroeneveld400 at gmail.com> wrote:
> > >> > Thanks Neil. That's what I meant. Sorry for the typo and confusing.
> > >> >
> > >> >
> > >> > On 31-08-14 21:01, Neil Sloane wrote:
> > >> >>
> > >> >> Arie said:
> > >> >>
> > >> >> It looks like that the list of partial sums of A164349is equal
> > toA10140.
> > >> >>
> > >> >> but more to the point, what he meant was:
> > >> >>
> > >> >> It looks like that the list of partial sums of A164349 is equal to
> > >> >> A101402,
> > >> >>
> > >> >> a very nice remark, since the latter is the sequence this
> discussion
> > >> >> is all about.
> > >> >>
> > >> >> But the discussion has gone off the boil - nothing for three days.
> > >> >> Benoit, can you wrap it up before the semester begins
> > >> >>   in a few days?
> > >> >>
> > >> >> Neil
> > >> >>
> > >> >> Neil
> > >> >>
> > >> >> On Thu, Aug 28, 2014 at 4:23 AM, Aai <agroeneveld400 at gmail.com>
> > wrote:
> > >> >>>
> > >> >>> Sorry. Premature sending.
> > >> >>>
> > >> >>>
> > >> >>>
> > >> >>> It looks like that the list of partial sums of A164349is equal
> > >> toA10140.
> > >> >>>
> > >> >>> A164349 comment
> > >> >>>
> > >> >>> The proportion of 0's in this sequence converges to a number close
> > to
> > >> >>> 0.645059.The constantsuggested by you is also
> > >> >>>
> > >> >>> 1 - 0.645059 = 0.354941
> > >> >>>
> > >> >>> the proportion of the number of 1's.
> > >> >>>
> > >> >>>
> > >> >>>
> > >> >>>
> > >> >>>
> > >> >>>
> > >> >>>
> > >> >>>> On 27-08-14 18:39, Charles Greathouse wrote:
> > >> >>>>>
> > >> >>>>> Sequence A101402 appears to be nearly linear. For the first
> 10,000
> > >> >>>>> terms
> > >> >>>>> there is a constant k such that |a(n) - kn| < 2 (e.g., take k =
> > >> 0.355).
> > >> >>>>> Can
> > >> >>>>> anyone prove or disprove that a(n) = kn + O(1) for some constant
> > k?
> > >> In
> > >> >>>>> the
> > >> >>>>> (likely?) latter case, can another reasonable bound be found,
> > maybe
> > >> >>>>> O(log
> > >> >>>>> n)? I can't even think of a technique that would work here.
> > >> >>>>>
> > >> >>>>> I just checked to a million and it looks like the same holds.
> > Here I
> > >> >>>>> used
> > >> >>>>> k
> > >> >>>>> = 0.3549419505. Probably going to 10 million would require
> > relaxing
> > >> the
> > >> >>>>> bound slightly; already by a million the choice of constant is
> > very
> > >> >>>>> constrained.
> > >> >>>>>
> > >> >>>>> Charles Greathouse
> > >> >>>>> Analyst/Programmer
> > >> >>>>> Case Western Reserve University
> > >> >>>>>
> > >> >>>>> _______________________________________________
> > >> >>>>>
> > >> >>>>> Seqfan Mailing list - http://list.seqfan.eu/
> > >> >>>>
> > >> >>>>
> > >> >>> --
> > >> >>> Met vriendelijke groet,
> > >> >>> @@i = Arie Groeneveld
> > >> >>>
> > >> >>>
> > >> >>> _______________________________________________
> > >> >>>
> > >> >>> Seqfan Mailing list - http://list.seqfan.eu/
> > >> >>
> > >> >>
> > >> >>
> > >> >
> > >> > --
> > >> > Met vriendelijke groet,
> > >> > @@i = Arie Groeneveld
> > >> >
> > >> >
> > >> > _______________________________________________
> > >> >
> > >> > Seqfan Mailing list - http://list.seqfan.eu/
> > >>
> > >> _______________________________________________
> > >>
> > >> Seqfan Mailing list - http://list.seqfan.eu/
> > >>
> > >
> > > _______________________________________________
> > >
> > > Seqfan Mailing list - http://list.seqfan.eu/
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

-- 
Dear Friends, I have now retired from AT&T. New coordinates:

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