[seqfan] Re: A006899, A108906 and similar sequences : divergence proof?

[jean-paul allouche]

Hi Bernard

Though it is not quite your question you might be interested, if you 
have not
seen that before, by what Hellegouarch wrote, see (in  French):

http://smf4.emath.fr/Publications/Gazette/1999/81/smf_gazette_81_25-39.pdf
http://smf4.emath.fr/Publications/Gazette/1999/82/smf_gazette_82_13-25.pdf
also see
http://www.etab.ac-caen.fr/cdgaulle/cpge/Maths/Math_et_musiqueAPM.pdf

best
jp

Le 25/05/14 00:29, Bernard Vatant a écrit :
> Merci Jean-Paul!
>
> Indeed this paper is definitely a step forward, I'll munch over it and see
> what I get to.
> I did not write in the first message where my motivation for that question
> came from.
> I was wondering about frequencies of harmonics, octaves and fifths, the
> fact that 2^7 and (3/2)^12 are different but close enough to allow at all
> the tuning of instruments, modulo arrangements with the wolf fifth,
> temperaments and the like. And I've been wondering if 2^n and (3/2)^k could
> come closer for higher values of k,n. Before getting to the widest
> conjecture of my first message, I was just wondering if the lowest bound of
> |2^n -(3/2)^k| was > 0, or if it could get arbitrarily small for great
> values of k,n. The paper seems at least to provide an answer to this first
> question.
>
> Best regards
>
> 2014-05-24 14:20 GMT+02:00 jean-paul allouche <jean-paul.allouche at imj-prg.fr
>> :
>> oops my answer left before I finished writing it
>> I wanted to add that the hypothesis a and b
>> multiplicatively independent (i.e., no positive
>> integer power of a is equal to an integer power
>> of b) has been omitted but is clearly necessary
>> in that corollary 1.8
>>
>> best
>>
>> jp
>>
>>
>> Le 24/05/14 01:26, Bernard Vatant a écrit :
>>
>>> Hello sequence fans
>>>
>>>
>>> I'm new to this list and the archives are huge, so please point me to any
>>> previous message if this has been answered.
>>> I've been struggling with sequences formed by merging two geometric
>>> progressions, the simpler being A006899, merging powers of 2 and 3, and
>>> particularly in the sequence of differences between its successive terms,
>>> such as A108906.
>>> Although it seems highly "obvious" by looking at the 1000 first terms
>>> of A108906 that it should be divergent, I could not find any proof of that
>>> divergence, nor even a statement that it is indeed divergent.
>>> I have a strong conjecture for any real numbers p and q such as 1 < p < q
>>> and p^k != q^n for all integers k,n
>>> Define the increasing sequence u(n) containing both powers of p or q on
>>> the
>>> model of A006899, and its differences sequence d(n)=u(n+1)-u(n) on the
>>> model of A108906.
>>> My conjecture is that d(n) is divergent ...
>>> I would be happy to have a proof for p and q integers, and even for p=2
>>> and
>>> q=3.
>>>
>>> Thanks for any clue, related works etc.
>>>
>>> Bernard
>>>
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