# Inspired By Garrett Clarke

For natural number $$p$$, let $$n^{p}$$ denote the $$p$$-th arithmetic derivative of a natural number $$n$$. Define $$A_{n}$$ as the set of all $$p$$-th derivatives of $$n$$. Find the number of all $$n<10^{10}$$ such that $$A_{n}$$ is a singleton set.

Details and Assumptions

• A singleton set is a set with a single element whose multiplicity doesn't matter, for example $$A=\{a\}$$ is a singleton set. Another example is $$S=\{e,e,e,e,e,e,e\}$$ and $$T=\{k,k,k,k,\ldots\}$$.
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