number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

, an unusual number is a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

''n'' whose largest

prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

is strictly greater than

\sqrt

. A ''k''-

smooth number In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × ...

has all its prime factors less than or equal to ''k'', therefore, an unusual number is non-

\sqrt

-smooth.

Relation to prime numbers

All

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

s are unusual. For any prime ''p'', its multiples less than ''p''² are unusual, that is ''p'', ... (''p''-1)''p'', which have a density 1/''p'' in the interval (''p'', ''p''²).

Examples

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 (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, ...

Distribution

If we denote the number of unusual numbers less than or equal to ''n'' by ''u''(''n'') then ''u''(''n'') behaves as follows:

Richard Schroeppel Richard C. Schroeppel (born 1948) is an American mathematician born in Illinois. His research has included magic squares, elliptic curves, and cryptography. In 1964, Schroeppel won first place in the United States among over 225,000 high school st ...

stated in 1972 that the asymptotic

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

that a randomly chosen number is unusual is ln(2). In other words: :

\lim_ \frac = \ln(2) = 0.693147 \dots\, .

External links

* {{Divisor classes Integer sequences