Arithmetic Derivative - Inequalities and Bounds

Inequalities and Bounds

E. J. Barbeau examined bounds of the arithmetic derivative. He found that the arithmetic derivative of natural numbers is bounded by

n' \leq \frac{n \log_k n}{k}

where k is the least prime in n and

n' \geq sn^{\frac{s-1}{s}}

where s is the number of prime factors in n. In both bounds above, equality occurs only if n is a perfect power of 2, that is for some m.

Alexander Loiko, Jonas Olsson and Niklas Dahl found that it is impossible to find similar bounds for the arithmetic derivative extended to rational numbers by proving that between any two rational numbers there are other rationals with arbitrary large or small derivatives.

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