Largest prime divisor: Difference between revisions

Revision as of 02:18, 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 19: / Line 19: @@
 {{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.