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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/220px-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 (composite) 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 the HAKMEM (1972), Item #29 [1] 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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Richard C. Schroeppel is an American mathematician born in Illinois. His research has included magic squares, elliptic curves, and cryptography. In 1964, Schroeppel won first place in the United States among over 225,000 high school students in the Annual High School Mathematics Examination, a contest sponsored by the Mathematical Association of America and the Society of Actuaries. In both 1966 and 1967, Schroeppel scored among the top 5 in the U.S. in the William Lowell Putnam Mathematical Competition. In 1973 he discovered that there are 275,305,224 normal magic squares of order 5. In 1998–1999 he designed the Hasty Pudding Cipher, which was a candidate for the Advanced Encryption Standard, and he is one of the designers of the SANDstorm hash, a submission to the NIST SHA-3 competition.
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