Kynea number

Given $n$, compute $4^n+2^{n+1}-1$ or ${(2^n+1)}^2-2$ or

$$4^n+\sum _{i=0}^n{2}^i.$$

Any of these formulas gives the Kynea number for $n$.

The first few Kynea numbers are 7, 23, 79, 287, 1087, 4223, 16639, 66047, 263167, 1050623, 4198399, 16785407, 67125247 (listed in A093069 of Sloane’s OEIS). Every third Kynea number is divisible by 7, thus prime Kynea numbers can’t have $n=3x+2$ (except of course for $n=2$. The largest Kynea number known to be prime is ${(2^{281621}+1)}^2-2$, found by Cletus Emmanuel in November of 2005, using k-Sieve from Phil Comody and OpenPFGW.