This notion of derivative of a number depends very weakly on the number itself. It only depends on the multiset of exponents in the number’s factorization into

This notion of derivative of a number depends very weakly on the number itself. It only depends on the multiset of exponents in the number’s factorization into a product of prime powers. In this way it’s like the divisor counting function, the omega and Omega functions, the möbius function, etc. As such, its value distribution might be interesting. Something analytic number theorists might like to play around with.