unusual number

An unusual number or $\sqrt{n}$-rough number $n$ 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).

Title	unusual number
Canonical name	UnusualNumber
Date of creation	2013-03-22 18:09:43
Last modified on	2013-03-22 18:09:43
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11A51