%N Arithmetic mean of successive Mersenne primes.
%C Sequence is based on the following (false) conjecture Shanks mentions in his paper (p.6), which someone else made: "If M_p1 and M_p2 are successive Mersenne primes, their arithmetic mean is also prime." The first four values in the sequence are prime, while the rest up to (2^756839-1 + 2^216091-1)/2 are not. Conjecture: there are no more primes in this sequence after 4159.
%H D. Shanks, <a href="http://cadigweb.ew.usna.edu/%7ewdj/mcmath/shanks_squfof.pdf">Squfof Notes</a>
%e a(3)=79 because (31+127)/2 = 79.
%Y Cf. A000043.
%A _Jason Earls_, Oct 24 2004