Largest prime divisor: Difference between revisions

Latest revision as of 02:26, 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

Initial values

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

Lower bound

There are infinitely many powers of two, and hence, $a (n) = 2$ for infinitely many numbers and this is the best lower bound.

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==
 {{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===
-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===
+{{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==