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 probability of learning something unusual from a newspaper is far greater than that of experiencing it; in other words, it is in the realm of the abstract that the more important things happen in these times, and it is the unimportant that happens in real life.”
—Robert Musil (1880–1942)
“The Oregon [matter] and the annexation of Texas are now all- important to the security and future peace and prosperity of our union, and I hope there are a sufficient number of pure American democrats to carry into effect the annexation of Texas and [extension of] our laws over Oregon. No temporizing policy or all is lost.”
—Andrew Jackson (1767–1845)