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).
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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

• Largest prime power divisor
• Square-free part