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 ${\text{'}}^{}$ 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: $1^{\prime }=0$ $p^{\prime }=1$ for any prime number $p$ Leibniz rule: $(ab{)}^{\prime }=a^{\prime }b+a{b}^{\prime }$ 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 $n^{\prime }$ is given by $n^{\prime }=n(\sum _{i=1}^r\frac{k_i}{p_i})$