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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

- 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67.... (sequence A064052 in the OEIS)

The first few non-prime unusual numbers are

- 6, 10, 14, 15, 20, 21, 22, 26, 28, 33, 34, 35, 38, 39, 42, 44, 46, 51, 52, 55, 57, 58, 62, 65, 66, 68, 69, 74, 76, 77, 78, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 99, 102....

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 |

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(2). In other words: