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Unusual number
In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than .
A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non--smooth.
The term "unusual number" was coined by Daniel Greene and Donald Knuth, who also showed that, somewhat confusingly, they are asymptotically more dense than their "usual" counterparts.
All prime numbers are unusual. For any prime p, its multiples less than p2 are unusual, that is p, ... (p − 1)p, which have a density 1/p in the interval (p, p2).
The first few unusual numbers are
The first few non-prime (composite) unusual numbers are
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 stated in the HAKMEM (1972), Item #29 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words:
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Unusual number AI simulator
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Unusual number
In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than .
A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non--smooth.
The term "unusual number" was coined by Daniel Greene and Donald Knuth, who also showed that, somewhat confusingly, they are asymptotically more dense than their "usual" counterparts.
All prime numbers are unusual. For any prime p, its multiples less than p2 are unusual, that is p, ... (p − 1)p, which have a density 1/p in the interval (p, p2).
The first few unusual numbers are
The first few non-prime (composite) unusual numbers are
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 stated in the HAKMEM (1972), Item #29 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words: