Largest prime divisor

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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


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.


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.

Relation with other arithmetic functions