Number derivative

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In mathematics, the number derivative can be defined for integers, based on prime factorization and in analogy with the product rule for the derivative of a function. For a natural number k, the number derivative k' is defined by

$k' = \begin{cases} 0, & \mbox{if } k = 0, 1\\ 1, & \mbox{if } k \mbox{ prime} \\ m'n + mn', & \mbox{if } k = mn \end{cases}$

The sequence of number derivatives for k = 0, 1, 2, ... begins (sequence A003415 in OEIS):

0, 0, 1, 1, 4, 1, 5, 1, 12, 6, 7, 1, 16, 1, 9, ...

Some important concepts in number theory can be related to equations of number derivatives. For example, Goldbach's conjecture would imply the existence of an n so that n' = 2k for every k, and the twin prime conjecture would imply that there are infinitely many k for which k'' = 1.