Unusual number
In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than square root|.
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, which have a density 1/p in the interval.Examples
The first few unusual numbers are
The first few non-prime unusual numbers areDistribution
If we denote the number of unusual numbers less than or equal to n by u then u behaves as follows:
| n | u | u / n |
| 10 | 6 | 0.6 |
| 100 | 67 | 0.67 |
| 1000 | 715 | 0.72 |
| 10000 | 7319 | 0.73 |
| 100000 | 73322 | 0.73 |
| 1000000 | 731660 | 0.73 |
| 10000000 | 7280266 | 0.73 |
| 100000000 | 72467077 | 0.72 |
| 1000000000 | 721578596 | 0.72 |
Richard Schroeppel stated in 1972 that the asymptotic probability that a randomly chosen number is unusual is ln. In other words:
