[seqfan] Re: Near-linear sequence

Charles Greathouse charles.greathouse at case.edu
Tue Sep 16 21:17:15 CEST 2014


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/
>
> _______________________________________________
>
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