Arithmetic derivative: Difference between revisions

Revision as of 00:33, 23 June 2012

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 $n'=n\left(\sum _{i=1}^{r}{\frac {k_{i}}{p_{i}}}\right)$

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 | using Leibniz rule and specification on primes || It is defined by the following three conditions:<br><math>1' = 0</math><br><math>p' = 1</math> for any [[prime number]] <math>p</math><br>Leibniz rule: <math>(ab)' = a'b + ab'</math> for any (possibly equal, possibly distinct) natural numbers <math>a,b</math>
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-| direct definition in terms of prime factorization || Consider a natural number <math>n</math> with prime factorization <math>n =  p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}</math> where the <math>p_i</math> are all distinct primes and the <math>k_i</math> are all ''positive'' integers (possibly repeated). Then the arithmetic derivative <math>n'</math> is given by <math>n' = n \left\sum_{i=1}^r \frac{k_i}{p_i}\right)</math>
+| direct definition in terms of prime factorization || Consider a natural number <math>n</math> with prime factorization <math>n =  p_1^{k_1}p_2^{k_2} \dots p_r^{k_r}</math> where the <math>p_i</math> are all distinct primes and the <math>k_i</math> are all ''positive'' integers (possibly repeated). Then the arithmetic derivative <math>n'</math> is given by <math>n' = n \left(\sum_{i=1}^r \frac{k_i}{p_i}\right)</math>
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