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 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
[edit]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
[edit]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 (composite) 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, ... (sequence A063763 in the OEIS)
Distribution
[edit]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:
References
[edit]- ^ Schroeppel, Richard (April 1995) [1972-02-29]. Baker, Henry Givens Jr. (ed.). "ITEM 29". HAKMEM (retyped & converted ed.). Cambridge, Massachusetts, USA: Artificial Intelligence Laboratory, Massachusetts Institute of Technology (MIT). AI Memo 239 Item 29. Archived from the original on 2024-02-24. Retrieved 2024-06-16.
External links
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