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.

Relation to prime numbers

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 are

Distribution

If we denote the number of unusual numbers less than or equal to n by u then u behaves as follows:
nuu / n
1060.6
100670.67
10007150.72
1000073190.73
100000733220.73
10000007316600.73
1000000072802660.73
100000000724670770.72
10000000007215785960.72

Richard Schroeppel stated in the HAKMEM, Item #29 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words: