Arithmetic derivative

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

The arithmetic derivative or number derivative is an arithmetic function, specifically a function from $\mathbb {N}$ to $\mathbb {N} _{0}$ denoted by the ${}^{'}$ superscript, defined in a number of equivalent ways.

Definition type	Definition details
using Leibniz rule and specification on primes	It is defined by the following three conditions: $1'=0$ $p'=1$ for any prime number $p$ Leibniz rule: $(ab)'=a'b+ab'$ for any (possibly equal, possibly distinct) natural numbers $a,b$
direct definition in terms of prime factorization	Consider a natural number $n$ with prime factorization $n=p_{1}^{k_{1}}p_{2}^{k_{2}}\dots p_{r}^{k_{r}}$ where the $p_{i}$ are all distinct primes and the $k_{i}$ are all positive integers (possibly repeated). Then the arithmetic derivative $n'$ is given by Failed to parse (syntax error): {\displaystyle n' = n \left\sum_{i=1}^r \frac{k_i}{p_i}\right)}