On the sum of digits of $n^k$ Reading another question on the sum of the digits of $2^n$ i started wondering whether there exist a $\alpha\in\mathbb{N}$ such that for every $n>\alpha$ we have $S(2^{n+1})>S(2^n)$, where $S(n)$ is the sum of the digits of $n$.
Via a computer program I plotted the graph of $S(2^n)$ for $n$ up to $10000$ and there clearly isn't a point in this interval after which $S(2^n)$ becomes increasing.
With the same computer program I plotted the graphs of $S(k^n)$ for several small $k$ and $n$ up to $10000$, with the same results (excluding $k=0,1$ or $10$, for which the sum of the digits is obviously constant).
edit:
Here is a graph of $S(2^n)$ for $1\le n\le 1000$, as you can see the values seems to grow (and this observation can be made also on the graph with higher values of $n$), but in a very oscillating manner

So, my question is, is there a $k\in\mathbb{N}$ such that $S(k^n), n\in\mathbb{N}$ eventually becomes increasing?
Later edit:
In this MO question is provided a reference for the fact that for any $n,s\in\Bbb{N}$ there exist only finitely many values of $k\in\Bbb{N}$ such that $S(n^k)<s$ so we know that $S(n^k)$ can be arbitratily big for any $n$ (of course $n$ must not be a power of $10$).
Even later edit:
Apparently the specific case of $S(2^n)$ had already been asked and negatively answered (thanks to Erick Wong for pointing it out, I somehow missed the question when i searched to see if mine was a duplicate).
I'm still interested in whether there is some $k$ such that $S(k^n)$ eventually becomes increasing
a few months older edit:
The question was reposted on MO
 A: Not an answer. Just a related plot I produced and feel like sharing.
This seems to very much support the model that in $2^n$ there are $n\log_{10}2$ random digits. In the figure below the blue cloud of dots gives the difference $S(n)-P(n)$, where $S(n)$ is the actual sum of base ten digits of $2^n$, and 
$P(n)=\frac92n\log_{10}2$ is the predicted sum of digits. The orange and red curves are the +1SD and +2SD curves respectively. The square of the difference of a random digit from $9/2$ has expected value $33/4$, so these curves are $k\sqrt{33 n \log_{10}2}/2$ with $k=1$ and $k=2$.
The range covered in the plot is $1\le n\le 10000$.

A: I would say that there is no such $\alpha$. In fact I would bet my house on it. It would be astonishing if (say) $S(2^n)$ were found to be eventually monotonic increasing. But I don't think anybody has a proof that it's not.
In the other direction: Does $S(2^n)$ even tend to infinity as $n \to \infty$? Again, it would be astonishing if it didn't. But I don't think anybody has a proof that it does.
