[seqfan] Re: Near-linear sequence

[Charles Greathouse]

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/
>
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