Claim :
For any even number $n$ there is at least one prime number of the form :
$p=k\cdot2^{n}-1$
with following properties : $k=2^{a-n}+1 , n\leq a < 2n , $ and $a,n\in \mathbf{Z^{+}}$
Can someone give me a counterexample ?
I have checked statement for each $n$ up to $n=100$ , but I am also aware of strong law of small numbers so this statement could be false.
EDIT:
One necessary condition is that there is at least one prime number on interval :
$[2^{n+1}-1,2^{2n-1}+2^{n}-1]$
One can easily show that this is true by using Bertrand's theorem.