Arithmetic derivative - Revision history

Vipul at 00:42, 23 June 2012

2012-06-23T00:42:01Z

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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>
 |}

Vipul: /* Definition */

2012-06-23T00:33:22Z

Definition

← Older revision		Revision as of 00:33, 23 June 2012
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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>		\| 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>
	\|-		\|-
	\| 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>
	\|}		\|}

Vipul: Created page with "{{arithmetic function}} ==Definition== The '''arithmetic derivative''' or '''number derivative''' is an arithmetic function, specifically a function from $\mathbb{N..."$

2012-06-23T00:33:08Z

Created page with "{{arithmetic function}} ==Definition== The '''arithmetic derivative''' or '''number derivative''' is an arithmetic function, specifically a function from <math>\mathbb{N..."

New page

{{arithmetic function}}

==Definition==

The '''arithmetic derivative''' or '''number derivative''' is an [[arithmetic function]], specifically a function from <math>\mathbb{N}</math> to <math>\mathbb{N}_0</math> denoted by the <math>{}^'</math> superscript, defined in a number of equivalent ways.

{| class="sortable" border="1"
! Definition type !! Definition details
|-
| 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>
|-
| 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>
|}

Arithmetic derivative - Revision history

Vipul at 00:42, 23 June 2012

Vipul: /* Definition */

Vipul: Created page with "{{arithmetic function}} ==Definition== The '''arithmetic derivative''' or '''number derivative''' is an arithmetic function, specifically a function from \mathbb{N..."

Vipul: Created page with "{{arithmetic function}} ==Definition== The '''arithmetic derivative''' or '''number derivative''' is an arithmetic function, specifically a function from $\mathbb{N..."$