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

[Bernard Vatant]

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