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

[jean-paul allouche]

Hi

May be the following paper 
http://pub.math.leidenuniv.nl/~evertsejh/dio2011-linforms.pdf
(see Corollary 1.8) can help
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
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/