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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 Sparsely totient number ： ウィキペディア英語版 Sparsely totient number In mathematics, a sparsely totient number is a certain kind of natural number. A natural number, ''n'', is sparsely totient if for all ''m'' > ''n'', :$\backslash varphi(m)>\backslash varphi(n)$ where $\backslash varphi$ is Euler's totient function. The first few sparsely totient numbers are: 2, 6, 12, 18, 30, 42, 60, 66, 90, 120, 126, 150, 210, 240, 270, 330, 420, 462, 510, 630, 660, 690, 840, 870, 1050, 1260, 1320, 1470, 1680, 1890, ... . For example, 18 is a sparsely totient number because ϕ(18) = 6, and any number ''m'' > 18 falls into at least one of the following classes: #''m'' has a prime factor ''p'' ≥ 11, so ϕ(''m'') ≥ ϕ(11) = 10 > ϕ(18). #''m'' is a multiple of 7 and ''m''/7 ≥ 3, so ϕ(''m'') ≥ 2ϕ(7) = 12 > ϕ(18). #''m'' is a multiple of 5 and ''m''/5 ≥ 4, so ϕ(''m'') ≥ 2ϕ(5) = 8 > ϕ(18). #''m'' is a multiple of 3 and ''m''/3 ≥ 7, so ϕ(''m'') ≥ 4ϕ(3) = 8 > ϕ(18). #''m'' is a power of 2 and ''m'' ≥ 32, so ϕ(''m'') ≥ ϕ(32) = 16 > ϕ(18). The concept was introduced by David Masser and Peter Shiu in 1986. As they showed, every primorial is sparsely totient. ==Properties== * If ''P''(''n'') is the largest prime factor of ''n'', then $\backslash liminf\; P(n)/\backslash log\; n=1$. * $P(n)\backslash ll\; \backslash log^\backslash delta\; n$ holds for an exponent $\backslash delta=37/20$. * It is conjectured that $\backslash limsup\; P(n)\; /\; \backslash log\; n\; =\; 2$. 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「Sparsely totient number」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.