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:
“Only in the most unusual cases is it useful to determine whether a book is good or bad; for it is just as rare for it to be one or the other. It is usually both.”
—Robert Musil (1880–1942)
“Black lady,
what will I do
without your two flowers?
I have inhabited you, number by number.
I have pushed you in and out like a needle.”
—Anne Sexton (1928–1974)