Largest prime divisor: Difference between revisions
Line 10: | Line 10: | ||
{{oeis|A006530}} | {{oeis|A006530}} | ||
===Initial values=== | |||
The initial values of the largest prime divisor are: <math>1,2,3,2,5,3,7,2,3,5,11,3,13,7,5,2,17,3,19,5,7,11,23,\dots</math>. | |||
===Lower bound=== | ===Lower bound=== | ||
There are infinitely many powers of two, and hence, <math>a(n) = 2</math> for infinitely many numbers and this is the best lower bound. | |||
===Density results=== | ===Density results=== |
Latest revision as of 02:26, 9 February 2010
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 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.