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



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