Problem: Let $b \ge 2$ be a fixed integer,
and let $s_b(n)$ denote the sum of the base-$b$ digits of $n$.
Show that there are infinitely many positive
integers that cannot be represented in the from $n + s_b(n)$
where $n$ is a positive integer.

Solution: For brevity let $f(n) = n + s_b(n)$.
Select any integer $M$.
Observe that $f(x) \ge b^{2M}$ for any $x \ge b^{2M}$,
but also $f(b^{2M}-k) \ge b^{2M}$ for $k = 1, 2, \dots, M$,
since the base-$b$ expansion of $b^{2M}-k$ will start out with
at least $M$ digits $b-1$.

Thus $f$ omits at least $M$ values in $[1, b^{2M}]$ for any $M$.