Write $A_{10}(k)$ for the average of the base-10 digits of a positive integer $k$:

$A_{10}(k):=\tfrac{1}{L+1}(d_0+\dots+d_L)$, where $k=\sum_{i=0}^L d_i 10^i$ with $d_i\in\{0,\dots,9\}$

I wonder if there is a sequence $\{a_n\}$ of positive integers for which any non-obvious facts about the behaviour of the digit-average sequence $\{A_{10}(a_n)\}\subset[0,9]$ are known?

For example, do we know that $\liminf_{n\rightarrow\infty} A_{10}(2^n)>0$ ?