[seqfan] max(c) : (q + d) c^k <= q (c + 1)^k

[hv at crypt.org]

I'm revisiting A085098 (number of solutions to prod (a_i+1)/a_i = 2)
in the hope that with a more powerful computer and a bit more knowledge
I may have a better chance to find a(8).

One of the core calculations in my implementation is to find:
  max(c) : (q + d) c^k <= q (c + 1)^k
with c, d, q, k all in Z+, d <= q <= 2^(2^n), 3 <= k <= n.

My code to calculate that was pretty slow and complex, but it occurs
to me now that I should expect c to be close to c' = kq/d, and trying
that with the actual values involved in calculating a(3) to a(7) I find
c - c' always in {0, 1, 2}.

Can anyone put bounds on how far off c' could be from the true value
for n=8, or suggest an alternative way to calculate the true value
efficiently?

Hugo