Are there any explicit formula for a sum of digits for a power in the given base? A problem to be specific: find a sum of digits for a number $2^{100}$ in the system with a base 5. In the system with a base 10?
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$\begingroup$ What do you mean by "explicit formula"? $\endgroup$– j.c.Commented Oct 28, 2015 at 6:48
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$\begingroup$ There is an explicit formula for the digit sum of $a^b\bmod c$ if $a^b<10c$. $\endgroup$– Anthony QuasCommented Oct 28, 2015 at 6:52
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The $b^k$'s digit of the base $b$ expansion of $x$ is $\lfloor x/b^k \rfloor \mod b$ (where I'm using "mod" as a function rather than a relation). I doubt that you'll get anything much more "explicit" than $\sum_{k \ge 0} \left(\lfloor x/b^k \rfloor \mod b\right)$.
answered Oct 28, 2015 at 7:12