[math-fun] parity of floor(2^n / n)

What is known about the parity of floor of 2^n / n , for general values of n? Some information is given in the OEIS entry for the sequence A071354. Is it believed, or known, that as N approaches infinity, the number of n between 1 and N for which the floor of 2^n / n is even, divided by N, converges to 1/2 ? (This may be asking way too much: I don't even know how to prove that the floor of 2^n / n is odd for infinitely many values of n.) The entry in the OEIS mentions that the parity is always even when n is prime; this is a nice little application of the binomial theorem. E.g., for n = 5, (1+1)^5 = 1+5+10+10+5+1, so the floor of 2^5 / 5 is 1+2+2+1, which is even. Jim Propp