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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
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.
Relation to prime numbers
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).
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 non-prime 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
|
0.72
|
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:
