In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than (sequence A064052 in OEIS). All prime numbers are unusual.
A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non--smooth.
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....
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....
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 proved in 1972 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words:
Famous quotes containing the words unusual and/or number:
“The United States is unusual among the industrial democracies in the rigidity of the system of ideological control—”indoctrination” we might say—exercised through the mass media.”
—Noam Chomsky (b. 1928)
“A great number of the disappointments and mishaps of the troubled world are the direct result of literature and the allied arts. It is our belief that no human being who devotes his life and energy to the manufacture of fantasies can be anything but fundamentally inadequate”
—Christopher Hampton (b. 1946)