Number of the divisors - Revision history

FloraC: Cleanup

2023-10-19T15:13:48Z

Cleanup

← Older revision		Revision as of 15:13, 19 October 2023
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	~~The '''number~~ of divisors~~''' of a number (<tt>''d(n)''</tt>) can be found from its prime factorization as the product of the by-one incremented exponents of all present prime bases.~~		{{Wikipedia\| Table of divisors }}

	* If the number ~~<math>n</math> has the [[prime factorization]]~~		The '''number of divisors''' {{w\|Divisor function\|d(''n'')}} of a number ''n'' can be found from its [[prime factorization]] as the product of the by-one incremented exponents of all present prime bases.
	~~:: <math>n = p_1^~~{~~e_1}\cdot p_2^~~{~~e_2~~}~~\dotsm p_r^{e_r~~}~~,</math>~~
	~~: then:<ref>G. H. Hardy, E. M. Wright:~~ ''~~An Introduction to the Theory of Numbers.~~'' ~~4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme 273, p. 239.</ref>~~
	~~:: <math>d(n) = (e_1+1)(e_2+1) \dotsm (e_r+1)</math>~~
	* for [[~~coprime~~]] ~~numbers <math>m</math> and <math>n</math> applies:~~
	~~:: <math>d(mn) = d(m)\cdot d(n)</math>~~

	== ~~Some~~ Examples ==		If the number ''n'' has the prime factorization

			<math>\displaystyle n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},</math>

			then:<ref>G. H. Hardy, E. M. Wright: ''An Introduction to the Theory of Numbers.'' 4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme 273, p. 239.</ref>

			<math>\displaystyle d(n) = (e_1 + 1)(e_2 + 1) \dotsm (e_r + 1)</math>

			For {{w\|Coprime integers\|coprime}} numbers ''m'' and ''n'' it follows that

			<math>\displaystyle d(mn) = d(m) \cdot d(n)</math>

			== Examples ==

	{\| class="wikitable"		{\| class="wikitable"
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	== External links ==		== External links ==

	* [https://en.wikipedia.org/wiki/Table_of_divisors Table of divisors - Wikipedia]
	* [https://en.wikipedia.org/wiki/Divisor_function Divisor function - Wikipedia]
	* {{OEIS\|A000005}}		* {{OEIS\|A000005}}

	[[Category:Math]]		[[Category:Math]]

Eliora at 21:49, 22 November 2022

2022-11-22T21:49:57Z

← Older revision		Revision as of 21:49, 22 November 2022
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	\| <math>d(30) = (1+1)(1+1)(1+1) = 8</math>		\| <math>d(30) = (1+1)(1+1)(1+1) = 8</math>
	\|}		\|}

			== See also ==

			* [[Highly composite equal division]]

	== References ==		== References ==

Xenwolf: just a note about an interesting function

2020-05-29T09:47:11Z

just a note about an interesting function

New page

The '''number of divisors''' of a number (<tt>''d(n)''</tt>) can be found from its prime factorization as the product of the by-one incremented exponents of all present prime bases.

* If the number <math>n</math> has the [[prime factorization]]
:: <math>n = p_1^{e_1}\cdot p_2^{e_2}\dotsm p_r^{e_r},</math>
: then:<ref>G. H. Hardy, E. M. Wright: ''An Introduction to the Theory of Numbers.'' 4. Edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1, Theoreme 273, p. 239.</ref>
:: <math>d(n) = (e_1+1)(e_2+1) \dotsm (e_r+1)</math>
* for [[coprime]] numbers <math>m</math> and <math>n</math> applies:
:: <math>d(mn) = d(m)\cdot d(n)</math>

== Some Examples ==

{| class="wikitable"
! Number
! Prime factorization
! Number of divisors
|-
| 8
| <math>8 = 2^3</math>
| <math>d(8) = (3+1) = 4</math>
|-
| 12
| <math>12 = 2^2 \cdot 3</math>
| <math>d(12) = (2+1)(1+1) = 6</math>
|-
| 30
| <math>30 = 2 \cdot 3 \cdot 5</math>
| <math>d(30) = (1+1)(1+1)(1+1) = 8</math>
|}

== References ==

<references/>

== External links ==

* [https://en.wikipedia.org/wiki/Table_of_divisors Table of divisors - Wikipedia]
* [https://en.wikipedia.org/wiki/Divisor_function Divisor function - Wikipedia]
* {{OEIS|A000005}}

[[Category:Math]]