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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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
The initial values of the largest prime divisor are: .
There are infinitely many powers of two, and hence, for infinitely many numbers and this is the best lower bound.
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.