Unusual number

In number theory, an unusual number is a natural number n whose largest prime factor is strictly greater than $\sqrt{n}$ (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-$\sqrt{n}$-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....

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).

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 stated in 1972 that the asymptotic probability that a randomly chosen number is unusual is ln(2). In other words:

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

External links

• Weisstein, Eric W., "Rough Number", MathWorld.