Assuming that $p$ is an odd prime. How many primes have the form $(2^p+1)/3$? Is the number finite? Mathematica calculation shows that there are 23 such primes when $p$ ranges over the first 500 primes. Here are those primes $p$,


3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 
313, 347, 701, 1709, 2617, 3539 


Originally, I am concerned with prime powers of the form $(2^m+1)/3$, where $m>0$ is an odd number. This number can be a prime power only when $m$ is an odd prime. According to a result of T. N. Shorrey and R. Tijdeman (Math. Scand. 39, 5-18 (1976)), there are only finite number of such primer powers that are not primes, though I still don't know how many. Mathematica calculation seems to suggest there are no such prime powers that are not primes.



How many primes have the form $(2^p+1)/3$?
I don't think anyone is going to answer this question, any more than anyone is going to decide the number of Mersenne primes any time soon, so for the sake of having an answer, I'll note that these numbers are tabulated at the [Online Encyclopedia of Integer Sequences](http://oeis.org/A000978) where they are called "Wagstaff numbers" and where many links and references are given. 


