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 $a(n)\le B$. 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 $1\le x\le 2$, the density of numbers $n$ such that $a(n)\ge n^{1/x}$ is given by $-\log x$. Hence the density of $n^{1/x}$-smooth numbers is $1-\log x$.
• For general $x\ge 1$, the density of numbers $n$ such that $a(n)\ge n^{1/x}$ 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

• Largest prime power divisor
• Square-free part