Largest prime divisor: Difference between revisions

Revision as of 01:13, 9 February 2010

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

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

For a positive real number $B$ , we say that $n$ is a $B$ -smooth number if $a (n) \leq B$ . Otherwise, $n$ is a $B$ -rough number.

Behavior

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A006530

Lower bound

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

Density results

Further information: Dickman-de Bruijn function

For $1 \leq x \leq 2$ , the density of numbers $n$ such that $a (n) \geq n^{1 / x}$ is given by $- \log x$ (hence the density of $n^{1 / x}$ -smooth numbers is $1 - \log x$ .
For general $x \geq 1$ , the density of numbers $n$ such that $a (n) \geq n^{1 / x}$ 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

@@ Line 3: / Line 3: @@
 ==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==
@@ Line 12: / Line 14: @@
 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==