Largest prime divisor

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