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不尋常數

不尋常數(英語:unusual number)是指一整數n的最大質因數大於,所有質數均為不尋常數。

k-光滑數是指其最大質因數小於或等於k,因此若整數n不是光滑數,此整數就是不尋常數。

若用u(n)表示小於等於n的整數中的不尋常數個數,u(n)和n有以下的關係:

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

數學家理查德·施羅培爾英语Richard Schroeppel在1972年證明了若任意選擇整數,選到不尋常數的漸進機率ln(2),也就是說:

例子

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前幾個不尋常數為:

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數列A064052

前幾個非質數的不尋常數為:

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數列A063763

外部連結

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不尋常數
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