{"id":1361,"date":"2016-12-28T23:57:06","date_gmt":"2016-12-28T14:57:06","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1361"},"modified":"2021-11-03T19:59:50","modified_gmt":"2021-11-03T10:59:50","slug":"htb201501","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb201501\/","title":{"rendered":"\u4e00\u6a4b\u59272015\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b.\r\n\\(n\\) \u4ee5\u4e0b\u306e\u6b63\u306e\u6574\u6570\u306e\u3046\u3061, \\(n\\) \u3068\u306e\u6700\u5927\u516c\u7d04\u6570\u304c \\(1\\) \u3068\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u3092 \\(E(n)\\) \u3067\u8868\u3059. \u305f\u3068\u3048\u3070\r\n\\[\r\nE(2) = 1 , \\ E(3) = 2 , \\ E(4) = 2 , \\ \\cdots , \\quad E(10) = 4 , \\ \\cdots\r\n\\]\r\n\u3067\u3042\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(E(1024)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(E(2015)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(m\\) \u3092\u6b63\u306e\u6574\u6570\u3068\u3057, \\(p\\) \u3068 \\(q\\) \u3092\u7570\u306a\u308b\u7d20\u6570\u3068\u3059\u308b. \\(n = p^m q^m\\) \u306e\u3068\u304d \\(\\dfrac{E(n)}{n} \\geqq \\dfrac{1}{3}\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\(1024 = 2^{10}\\) \u306a\u306e\u3067, \\(2\\) \u3092\u7d20\u56e0\u6570\u306b\u3082\u305f\u306a\u3044\u6574\u6570\u3092\u6570\u3048\u308c\u3070\u3088\u3044\u304b\u3089\r\n\\[\r\nE(1024) = 1024 -\\dfrac{1024}{2} = \\underline{512}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    \\(2015 = 5 \\cdot 13 \\cdot 31\\) \u306a\u306e\u3067, \\(5 , 13 , 31\\) \u306e\u3044\u305a\u308c\u3082\u7d20\u56e0\u6570\u306b\u3082\u305f\u306a\u3044\u6574\u6570\u3092\u6570\u3048\u308c\u3070\u3088\u3044\u304b\u3089\r\n\\[\\begin{align}\r\nE(2015) & = 2015 -\\dfrac{2015}{5} -\\dfrac{2015}{13} -\\dfrac{2015}{31} \\\\\r\n& \\quad +\\dfrac{2015}{5 \\cdot 13} +\\dfrac{2015}{5 \\cdot 31} +\\dfrac{2015}{13 \\cdot 31} \\\\\r\n& \\qquad -\\dfrac{2015}{5 \\cdot 13 \\cdot 31} \\\\\r\n& = 2015 \\left( 1 -\\dfrac{1}{5} \\right) \\left( 1 -\\dfrac{1}{13} \\right) \\left( 1 -\\dfrac{1}{31} \\right) \\\\\r\n& = 4 \\cdot 12 \\cdot 30 \\\\\r\n& = \\underline{1440}\r\n\\end{align}\\]\r\n

    (3)<\/strong><\/p>\r\n

    \\(p \\lt q\\) \u3068\u4eee\u5b9a\u3057\u3066\u3088\u3044\u306e\u3067\r\n\\[\r\np \\geqq 2 , \\ q \\geqq 3 \\quad ... [1]\r\n\\]\r\n(2)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308c\u3070\r\n\\[\r\nE(n) = n \\left( 1 -\\dfrac{1}{p} \\right) \\left( 1 -\\dfrac{1}{q} \\right)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{E(n)}{n} & = \\left( 1 -\\dfrac{1}{p} \\right) \\left( 1 -\\dfrac{1}{q} \\right) \\\\\r\n& \\geqq \\dfrac{1}{2} \\cdot \\dfrac{2}{3} \\quad ( \\text{\u2235} \\ [1] \\ ) \\\\\r\n& = \\dfrac{1}{3}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u6574\u6570\u3068\u3059\u308b. \\(n\\) \u4ee5\u4e0b\u306e\u6b63\u306e\u6574\u6570\u306e\u3046\u3061, \\(n\\) \u3068\u306e\u6700\u5927\u516c\u7d04\u6570\u304c \\(1\\) \u3068\u306a\u308b\u3082\u306e\u306e\u500b\u6570\u3092 \\(E(n)\\) \u3067\u8868\u3059. \u305f\u3068\u3048\u3070 \\[ E(2) = 1 , \\ E( … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[136],"tags":[146,137],"class_list":["post-1361","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2015","tag-hitotsubashi","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1361","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=1361"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1361\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1361"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1361"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1361"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}