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:
“Every woman who vacates a place in the teachers’ ranks and enters an unusual line of work, does two excellent things: she makes room for someone waiting for a place and helps to open a new vocation for herself and other women.”
—Frances E. Willard (1839–1898)
“Cultivated labor drives out brute labor. An infinite number of shrewd men, in infinite years, have arrived at certain best and shortest ways of doing, and this accumulated skill in arts, cultures, harvestings, curings, manufactures, navigations, exchanges, constitutes the worth of our world to-day.”
—Ralph Waldo Emerson (1803–1882)