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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 \sqrt{n}.

A k-smooth number has all its prime factors less than or equal to k, therefore, an unusual number is non-\sqrt{n}-smooth.

Relation to prime numbers

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).

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:

10000000007215785960.72

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:

:\lim_{n \rightarrow \infty} \frac{u(n)}{n} = \ln(2) = 0.693147 \dots, .

References

References

  1. (April 1995). "ITEM 29". [[MIT AI Lab.
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