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

[Bernard Vatant]

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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