[seqfan] Re: Near-linear sequence

Benoît Jubin benoit.jubin at gmail.com
Tue Sep 16 21:07:56 CEST 2014


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/



More information about the SeqFan mailing list