Cody Problem 47843 involved the arithmetic derivative of integers. In particular, D(p) = 1 if p is prime and D(mn) = n D(m) + m D(n). Therefore, the arithmetic derivatives of 1, 2, 3, 4, 5, and 6 are 0, 1, 1, 4, 1, and 5, respectively.
One might then ask about solving arithmetic differential equations (ADEs). Because the study of differential equations often starts with solving dy/dx = y, let’s consider the analogous ADE D(n) = n. The definition of the arithmetic derivative shows that no prime can solve this equation, but the sample calculations above show that the first (i.e., m = 1) solution is 4.
Write a function to compute the mth solution to this ADE. Because the solutions become large quickly, return the logarithm of the solution.
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George Berken Augusto Mazzei Christian Schröder Mehdi Dehghan

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