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![Demonstration, with Cuisenaire rods, that the number 10 is an unusual number, its largest prime factor being 5, which is greater than [?]10 [?] 3.16 Unusual number Cuisenaire rods 10.png](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Unusual_number_Cuisenaire_rods_10.png/440px-Unusual_number_Cuisenaire_rods_10.png)
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 |
| 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:
Divisibility-based sets of integers | ||
|---|---|---|
| Overview |  | |
| Factorization forms | ||
| Constrained divisor sums | ||
| With many divisors | ||
| Aliquot sequence-related | ||
| Base-dependent | ||
| Other sets | ||
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