Odd man out

What is the sum of the first two positive odd integers \(n\) such that the \(n^\text{th}\) arithmetic derivative of \(n\) is non-zero?

For example, the third arithmetic derivative of \(n\), is given by \(((n')')'\).

Clarification: The arithmetic derivative of \(n\) is given by:

  • \(n' = 0\) for \(n\) = 1 or 0.
  • \(n'\) = 1 if \(n\) is prime.
  • \(n' = a'b + b'a\) where \(a>1\) and \(b>1\) are factors of \(n\).

Note: If \(n\) has multiple factors you can choose any pair to get the arithmetic derivative.

(Image courtesy of "Top Lead Generators")

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