Does Conway's PRIMEGAME generate all prime powers of 2? Most sites I have visited reading on this interesting topic state something along the lines


  "the only powers of two (other than 2 itself) that occur in this sequence are those with prime exponent" (MathWorld)


or


  "After 2, this sequence contains the following powers of 2: [...] which are the prime powers of 2." (Wikipedia)


These careful formulations would imply that the set of powers of 2 generated in the sequence is a subset of prime powers of 2.

However, the OEIS seems absolutely certain that the two sets are equal: http://oeis.org/A034785

This result is also cited on other sites I do not consider very reliable for exact wording, like
http://esolangs.org/wiki/Fractran.

Honestly, I have not understood the internal mechanics of PRIMEGAME enough to answer my own question yet. However, I think it makes a significant difference in the interestingness of PRIMEGAME. Why would sites like MathWorld not state the full fact?
Yes, PRIMEGAME outputs $2^k$ if and only if $k$ is prime.

Conway's original paper is well worth reading if you can track it down.  You can also find a very clear exposition in Richard Guy's paper Conway's prime producing machine (Mathematics Magazine 56(1):26–33, 1983), including the wonderful cartoon below. (Yes, that's Conway with the Alexander horns, referring to a famous drawing by Simon Fraser.)  Conway himself posted a concise proof on the math-fun mailing list.  There's also a brief explanation at the OEIS blog.