let $a$ be a positive integer such that $$S(a^n+n)=1+S(n)$$ for any sufficiently large $n$ if and only if $a$ is a power of $10$
where $S(n)$ is digit sum of a positive integer $n$
if $a$ is a power of $10$,let $a=10^k$,for large $n$,then $10^{kn}>n$,so It is clear $$S(a^n+n)=S(10^{kn}+n)=1+S(n)$$ But for other hand,if $S(a^n+n)=1+S(n)$ I can't prove $a$ is a power of $10$