Roughness length

2013-12-03T23:44:42Z

129.15.109.254:

In [[number theory]], an '''unusual number''' is a [[natural number]] ''n'' whose largest [[prime factor]] is strictly greater than [[square root|<math>\sqrt{n}</math>]] {{OEIS|id=A064052}}. All [[prime number]]s are unusual.

A ''k''-[[smooth number]] has all its prime factors less than or equal to ''k'', therefore, an unusual number is non-<math>\sqrt{n}</math>-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....

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]] [[mathematical proof|proved]] in 1972 that the asymptotic [[probability]] that a randomly chosen number is unusual is [[Natural logarithm of 2|ln(2)]]. In other words:

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

== External links ==
* {{MathWorld|urlname=RoughNumber|title=Rough Number}}

{{Divisor classes}}

[[Category:Integer sequences]]

{{numtheory-stub}}

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Roughness length