# 2018-09 Sum of digits

For a positive integer $$n$$, let $$S(n)$$ be the sum of all decimal digits in $$n$$, i.e., if $$n = n_1 n_2 \dots n_m$$ is the decimal expansion of $$n$$, then $$S(n) = n_1 + n_2 + \dots + n_m$$. Find all positive integers $$n$$ and $$r$$ such that $$(S(n))^r = S(n^r)$$.

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Does there exist infinitely many positive integers $$n$$ such that the first digit of $$2^n$$ is $$9$$?