From A033844 it can be seen that prime(2^(n+1)) ~ 2*prime(2^n), which is explained by the prime number theorem. This led to the definition of A197072 (difference of these two expressions), which has again the property that A197072(n+1) ~ 2*A197072(n). This motivated the present sequence. (There is still a slow convergence to 2 of the ratio between successive terms here, e.g. a(55)/a(54) ~ 1.926.)