# Inspired By Garrett Clarke

**Number Theory**Level 5

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 \(T=\{k,k,k,k,\ldots\}\).