[math-fun] Sums of digits of powers of 2

Let E(n) and V(n) be the expected value and variance, respectively, of the sum of n independent draws from {0,1,2,...,9}; we can easily write E(n) and V(n) in closed form, and thereby extend to E(x) and V(x) where x is any positive real number. Let S(n) be the sum of the digits of 2^n, and let c = log_10 2, so that 2^n has about cn digits. What if anything is known (empirically or rigorously) about the distribution and autocorrelations of the sequence D_n whose nth term is (S(n)-E(cn))/sqrt(V(cn))? If that's obscure, just think of it as the sum of the digits of 2^n, adjusted to make it look like a Gaussian of mean 0 and variance 1. (The "D" is for discrepancy.) Jim Propp