Unusual number

Demonstration, with Cuisenaire rods, that the number 10 is an unusual number, its largest prime factor being 5, which is greater than √10 ≈ 3.16

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.... (sequence A064052 in the OEIS)

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....

Distribution

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.

Divisibility-based sets of integers

Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic Arithmetic number

Factorization forms	Prime Composite Semiprime Pronic Sphenic Square-free Powerful Perfect power Achilles Smooth Regular Rough Unusual

Constrained divisor sums	Perfect Almost perfect Quasiperfect Multiply perfect Hemiperfect Hyperperfect Superperfect Unitary perfect Semiperfect Practical Erdős–Nicolas

With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird

Aliquot sequence-related	Untouchable Amicable Sociable Betrothed

Other sets	Deficient Friendly Solitary Sublime Harmonic divisor Frugal Equidigital Extravagant

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