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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Golomb–Dickman constant ： ウィキペディア英語版 Golomb–Dickman constant In mathematics, the Golomb–Dickman constant arises in the theory of random permutations and in number theory. Its value is :$\backslash lambda\; =\; 0.62432\; 99885\; 43550\; 87099\; 29363\; 83100\; 83724\backslash dots.$ Let ''a''''n'' be the average — taken over all permutations of a set of size ''n'' — of the length of the longest cycle in each permutation. Then the Golomb–Dickman constant is : $\backslash lambda\; =\; \backslash lim\_\; \backslash frac.$ In the language of probability theory, $\backslash lambda\; n$ is asymptotically the expected length of the longest cycle in a uniformly distributed random permutation of a set of size ''n''. In number theory, the Golomb–Dickman constant appears in connection with the average size of the largest prime factor of an integer. More precisely, :$\backslash lambda\; =\; \backslash lim\_\; \backslash frac1n\; \backslash sum\_^n\; \backslash frac,$ where $P\_1(k)$ is the largest prime factor of ''k''. So if ''k'' is a ''d'' digit integer, then $\backslash lambda\; d$ is the asymptotic average number of digits of the largest prime factor of ''k''. The Golomb–Dickman constant appears in number theory in a different way. What is the probability that second largest prime factor of ''n'' is smaller than the square root of the largest prime factor of ''n''? Asymptotically, this probability is $\backslash lambda$. More precisely, :$\backslash lambda\; =\; \backslash lim\_\; \backslash text\backslash left\backslash$ where $P\_2(n)$ is the second largest prime factor ''n''. There are several expressions for $\backslash lambda$. Namely, :$\backslash lambda\; =\; \backslash int\_0^\backslash infty\; e^\; dt$ where $E\_1(t)$ is the exponential integral, :$\backslash lambda\; =\; \backslash int\_0^\backslash infty\; \backslash frac\; dt$ and :$\backslash lambda\; =\; \backslash int\_0^\backslash infty\; \backslash frac\; dt$ where $\backslash rho(t)$ is the Dickman function. == See also == * Random permutation * Random permutation statistics 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Golomb–Dickman constant」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.