# unusual number

An or $\sqrt{n}$ $n$ is an integer with a greatest prime factor exceeding $\sqrt{n}$. For example, the greatest prime factor of 102 is 17, which is greater than 11 (the square root of 102 rounded up to the next higher integer). 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, etc., listed in A064052 of Sloane’s OEIS. In fact, Donald Knuth and Donald Greene, who coined the term “unusual number,” remark that these numbers occur so frequently they’re not all that unusual. Unusual numbers include all of the prime numbers and many composites. Richard Schroeppel proved in HAKMEM 239 that the probability that a random integer is unusual is $\log 2$ (about 0.69314718).

## References

• 1 Donald Greene & Donald Knuth, Mathematics for the Analysis of Algorithms, 3rd edition. Boston: Birkhäuser (1990): 95 - 98
Title unusual number UnusualNumber 2013-03-22 18:09:43 2013-03-22 18:09:43 PrimeFan (13766) PrimeFan (13766) 4 PrimeFan (13766) Definition msc 11A51