Let's observe following number :
$ 4517\cdot 2^{332192811}+1$
I have noticed :
If $k\cdot 2^{2n+1}+1$ is prime number then $\gcd(k-1,3)=1$ , where $k,n \in Z^{+}$ , so
$\gcd(k-1,3)=1$ should be a necessary condition.
Since $\gcd(4516,3)=1$ this condition is fulfilled.
Proth weight of coefficient $4517$ is $w \approx 0.98199 $ which may be considered to be high value.
I would like to know are there some other necessary conditions that this number has to satisfy , so that might be considered like candidate for prime number ?