[seqfan] Natural numbers of the form (2^m + 3)/(2^k - 9)

[Hagen von Eitzen]

Dear seqfans,

I'm contemplating natural numbers of the form (2^m + 3)/(2^k - 9) and 
would like to hear some opinions:
Does their list start
1, 5, 37, 149, 293, 2341, 16981, 18725, 149797, 1198373, 9586981, 
76695845, 156180629, ...
or am I missing a few values?
Most of the numbers above occur in A046636, which corresponds to k=4, m 
= 2 mod 3.
Other subsequences come from k=6, m = 13 mod 20 and k=8, m = 22 mod 36.
There is no solution with k=10, the solutions with k=12, m = 336 mod 660 
are already quite large, but that does not rule out that some big-k 
solution might be rather small -  or does it?
I checked up to k=31 that there is no solution if k is odd. Can this be 
proved generally?

Thanks for your interest
Hagen