Unusual number

File:Unusual number Cuisenaire rods 10.png, 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, ... {{OEIS|id=A064052}}

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, ... {{OEIS|id=A063763}}

Distribution

If we denote the number of unusual numbers less than or equal to n by u(n) then u(n) behaves as follows:

class="wikitable"

|u(n)

|u(n) / n

|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

Richard Schroeppel stated in the HAKMEM (1972), Item #29{{cite journal |author-last=Schroeppel |author-first=Richard |author-link=Richard Schroeppel |editor-first=Henry Givens Jr. |editor-last=Baker |editor-link=Henry Baker (computer scientist) |title=ITEM 29 |journal=HAKMEM |date=April 1995 |orig-date=1972-02-29 |id=AI Memo 239 Item 29 |publisher=Artificial Intelligence Laboratory, Massachusetts Institute of Technology (MIT) |location=Cambridge, Massachusetts, USA |edition=retyped & converted |url=http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item29 |access-date=2024-06-16 |url-status=live |archive-url=https://web.archive.org/web/20240224184435/http://www.inwap.com/pdp10/hbaker/hakmem/number.html#item29 |archive-date=2024-02-24}} 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

External links

{{MathWorld|urlname=RoughNumber|title=Rough Number}}

Category:Integer sequences