In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than .
A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non--smooth.
All prime numbers are unusual. For any prime p, its multiples less than p² are unusual, that is p, ... (p-1)p, which have a density 1/p in the interval (p,p²).
The first few unusual numbers are
The first few non-prime unusual numbers are
If we denote the number of unusual numbers less than or equal to n by u(n) then u(n) behaves as follows:
|n||u(n)||u(n) / n|
Richard Schroeppel stated in 1972 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words: