# 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  be a natural number greater than . The largest prime divisor of  is defined as the largest among the primes  that divide . This is denoted as . By fiat, we set .

For a positive real number , we say that  is a -smooth number if . Otherwise,  is a -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: .

### Lower bound

There are infinitely many powers of two, and hence,  for infinitely many numbers and this is the best lower bound.

### Density results

Further information: Dickman-de Bruijn function

• For , the density of numbers  such that  is given by . Hence the density of -smooth numbers is .
• For general , the density of numbers  such that  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.