Unusual number
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In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than .
A ksmooth number has all its prime factors less than or equal to k, therefore, an unusual number is nonsmooth.
Relation to prime numbers
All prime numbers are unusual. For any prime p, its multiples less than p² are unusual, that is p, ... (p1)p, which have a density 1/p in the interval (p,p²).
Examples
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 nonprime 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....
Distribution
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  0.715 
10000  7319  0.7319 
100000  70128  0.70128 
Richard Schroeppel stated in 1972 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words:
External links
 Weisstein, Eric W. "Rough Number". MathWorld.
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