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.

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.^[1]

[1]

n	u(n)	u(n) / n
10	6	0.6
100	67	0.67
1000	715	0.72
10000	7319	0.73
100000	73322	0.73
1000000	731660	0.73
10000000	7280266	0.73
100000000	72467077	0.72
1000000000	721578596	0.72

v t e Divisibility-based sets of integers
Overview	Integer factorization Divisor Unitary divisor Divisor function Prime factor Fundamental theorem of arithmetic
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 Descartes Erdős–Nicolas
With many divisors	Abundant Primitive abundant Highly abundant Superabundant Colossally abundant Highly composite Superior highly composite Weird
Aliquot sequence-related	Untouchable Amicable (Triple) Sociable Betrothed
Base-dependent	Equidigital Extravagant Frugal Harshad Polydivisible Smith
Other sets	Arithmetic Deficient Friendly Solitary Sublime Harmonic divisor Refactorable Superperfect

Unusual number

Examples

Distribution

References

External links

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