Largest prime divisor - Revision history

Vipul: /* Behavior */

2010-02-09T02:26:23Z

Behavior

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	{{oeis\|A006530}}		{{oeis\|A006530}}

			===Initial values===

			The initial values of the largest prime divisor are: <math>1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3,19,5,7,11,23,\dots</math>.

	===Lower bound===		===Lower bound===

	~~There is no lower bound on the largest prime divisor of <math>n</math> as a function of <math>n</math>.~~ There are infinitely many powers of two, and hence, <math>a(n) = 2</math> for infinitely many numbers.		There are infinitely many powers of two, and hence, <math>a(n) = 2</math> for infinitely many numbers and this is the best lower bound.

	===Density results===		===Density results===

Vipul: /* Behavior */

2010-02-09T02:18:07Z

Behavior

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	{{further\|[[Dickman-de Bruijn function]]}}		{{further\|[[Dickman-de Bruijn function]]}}

	* For <math>1 \le x \le 2</math>, the density of numbers <math>n</math> such that <math>a(n) \ge n^{1/x}</math> is given by <math>-\log x</math> ~~(hence~~ the density of <math>n^{1/x}</math>-smooth numbers is <math>1 - \log x</math>.		* For <math>1 \le x \le 2</math>, the density of numbers <math>n</math> such that <math>a(n) \ge n^{1/x}</math> is given by <math>-\log x</math>. Hence the density of <math>n^{1/x}</math>-smooth numbers is <math>1 - \log x</math>.
	* For general <math>x \ge 1</math>, the density of numbers <math>n</math> such that <math>a(n) \ge n^{1/x}</math> is still positive. This density (or rather, the density of the complement) is described by the [[Dickman-de Bruijn function]], which occurs as the solution to a delay differential equation.		* For general <math>x \ge 1</math>, the density of numbers <math>n</math> such that <math>a(n) \ge n^{1/x}</math> is still positive. This density (or rather, the density of the complement) is described by the [[Dickman-de Bruijn function]], which occurs as the solution to a delay differential equation.

Vipul at 01:13, 9 February 2010

2010-02-09T01:13:28Z

← Older revision		Revision as of 01:13, 9 February 2010
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	==Definition==		==Definition==

	Let <math>n</math> be a [[natural number]] greater than <math>1</math>. The ''largest prime divisor''' of <math>n</math> is defined as the largest among the primes <math>p</math> that divide <math>n</math>. This is denoted as <math>a(n)</math>. By fiat, we set <math>a(1) = 1</math>.		Let <math>n</math> be a [[natural number]] greater than <math>1</math>. The '''largest prime divisor''' of <math>n</math> is defined as the largest among the primes <math>p</math> that divide <math>n</math>. This is denoted as <math>a(n)</math>. By fiat, we set <math>a(1) = 1</math>.

			For a positive real number <math>B</math>, we say that <math>n</math> is a <math>B</math>-[[smooth number]] if <math>a(n) \le B</math>. Otherwise, <math>n</math> is a <math>B</math>-''rough'' number.

	==Behavior==		==Behavior==
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	There is no lower bound on the largest prime divisor of <math>n</math> as a function of <math>n</math>. There are infinitely many powers of two, and hence, <math>a(n) = 2</math> for infinitely many numbers.		There is no lower bound on the largest prime divisor of <math>n</math> as a function of <math>n</math>. There are infinitely many powers of two, and hence, <math>a(n) = 2</math> for infinitely many numbers.

			===Density results===

			{{further\|[[Dickman-de Bruijn function]]}}

			* For <math>1 \le x \le 2</math>, the density of numbers <math>n</math> such that <math>a(n) \ge n^{1/x}</math> is given by <math>-\log x</math> (hence the density of <math>n^{1/x}</math>-smooth numbers is <math>1 - \log x</math>.
			* For general <math>x \ge 1</math>, the density of numbers <math>n</math> such that <math>a(n) \ge n^{1/x}</math> is still positive. This density (or rather, the density of the complement) is described by the [[Dickman-de Bruijn function]], which occurs as the solution to a delay differential equation.

	==Relation with other arithmetic functions==		==Relation with other arithmetic functions==

Vipul: /* Relation with other arithmetic functions */

2009-04-29T02:09:58Z

Relation with other arithmetic functions

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	* [[Largest prime power divisor]]		* [[Largest prime power divisor]]
	* [[~~Largest square~~-free ~~divisor~~]]		* [[Square-free part]]

Vipul: Created page with '{{arithmetic function}} ==Definition== Let $n$ be a natural number greater than $1$ . The ''largest prime divisor''' of $n$ is defined as th...'

2009-04-28T23:03:39Z

Created page with '{{arithmetic function}} ==Definition== Let <math>n</math> be a natural number greater than <math>1</math>. The ''largest prime divisor''' of <math>n</math> is defined as th...'

New page

{{arithmetic function}}

==Definition==

Let <math>n</math> be a [[natural number]] greater than <math>1</math>. The ''largest prime divisor''' of <math>n</math> is defined as the largest among the primes <math>p</math> that divide <math>n</math>. This is denoted as <math>a(n)</math>. By fiat, we set <math>a(1) = 1</math>.

==Behavior==

{{oeis|A006530}}

===Lower bound===

There is no lower bound on the largest prime divisor of <math>n</math> as a function of <math>n</math>. There are infinitely many powers of two, and hence, <math>a(n) = 2</math> for infinitely many numbers.

==Relation with other arithmetic functions==

* [[Largest prime power divisor]]
* [[Largest square-free divisor]]

Largest prime divisor - Revision history

Vipul: /* Behavior */

Vipul: /* Behavior */

Vipul at 01:13, 9 February 2010

Vipul: /* Relation with other arithmetic functions */

Vipul: Created page with '{{arithmetic function}} ==Definition== Let n be a natural number greater than 1. The ''largest prime divisor''' of n is defined as th...'

Vipul: Created page with '{{arithmetic function}} ==Definition== Let $n$ be a natural number greater than $1$ . The ''largest prime divisor''' of $n$ is defined as th...'