# [seqfan] Re: Near-linear sequence

Benoît Jubin benoit.jubin at gmail.com
Mon Sep 8 19:26:40 CEST 2014

```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
>>
>> 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/

```