{"id":1150,"date":"2015-07-02T22:23:07","date_gmt":"2015-07-02T13:23:07","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1150"},"modified":"2021-09-15T07:25:24","modified_gmt":"2021-09-14T22:25:24","slug":"ngr201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201404\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2014\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u8ca0\u3067\u306a\u3044\u6574\u6570 \\(N\\) \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d, \\(a _ 1 = N\\) , \\(a _ {n+1} = \\left[ \\dfrac{N}{2} \\right] \\quad ( n = 1, 2, 3, \\cdots )\\) \u3068\u3057\u3066\u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u3092\u5b9a\u3081\u308b. \u305f\u3060\u3057 \\([a]\\) \u306f, \u5b9f\u6570 \\(a\\) \u306e\u6574\u6570\u90e8\u5206\uff08 \\(k \\leqq a \\lt k+1\\) \u3068\u306a\u308b\u6574\u6570 \\(k\\) \uff09\u3092\u8868\u3059.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a _ 3 = 1\\) \u3068\u306a\u308b\u3088\u3046\u306a \\(N\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(0 \\leqq N \\lt 2^{10}\\) \u3092\u307f\u305f\u3059\u6574\u6570 \\(N\\) \u306e\u3046\u3061\u3067, \\(N\\) \u304b\u3089\u5b9a\u307e\u308b\u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u306e\u3042\u308b\u9805\u304c \\(2\\) \u3068\u306a\u308b\u3088\u3046\u306a\u3082\u306e\u306f\u3044\u304f\u3064\u3042\u308b\u304b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(0\\) \u304b\u3089 \\(2^{100} -1\\) \u307e\u3067\u306e \\(2^{100}\\) \u500b\u306e\u6574\u6570\u304b\u3089\u7b49\u3057\u3044\u78ba\u7387\u3067 \\(N\\) \u3092\u9078\u3073, \u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u3092\u5b9a\u3081\u308b. \u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u307f\u305f\u3059\u6700\u5c0f\u306e\u6b63\u306e\u6574\u6570 \\(m\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n

      \r\n
    1. (\uff0a)\u3000\u6570\u5217 \\(\\left\\{ a _ n \\right\\}\\) \u306e\u3042\u308b\u9805\u304c \\(m\\) \u3068\u306a\u308b\u78ba\u7387\u304c \\(\\dfrac{1}{100}\\) \u4ee5\u4e0b\u3068\u306a\u308b.<\/li>\r\n<\/ol><\/li>\r\n<\/ol>\r\n
      \r\n\r\n

      \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

      \\(a _ 3 = 1\\) \u3068\u306a\u308b\u306e\u306f\r\n\\[\r\na _ 2 = 2 ,3 \\ .\r\n\\]\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(a _ 2 = 2\\) \u3068\u306a\u308b\u306e\u306f\r\n\\[\r\na _ 1 = 4 , 5 \\ .\r\n\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(a _ 2 = 3\\) \u3068\u306a\u308b\u306e\u306f\r\n\\[\r\na _ 1 = 6 , 7 \\ .\r\n\\]<\/li>\r\n<\/ol>\r\n

        \u3088\u3063\u3066, \u6c42\u3081\u308b \\(N\\) \u306f\r\n\\[\r\nN = \\underline{4 , 5 , 6 , 7} \\ .\r\n\\]\r\n

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

        \u6761\u4ef6\u3092\u307f\u305f\u3059\uff08 \\(a _ n = m\\) \u3068\u306a\u308b\u9805\u304c\u3042\u308b \uff09 \\(N\\) \u306e\u500b\u6570\u3092 \\(I _ m\\) \u3068\u304a\u304f.
        \r\n\\(2^k \\leqq m \\lt 2^{k+1}\\) \u3092\u307f\u305f\u3059\u81ea\u7136\u6570 \\(m\\) \u306b\u5bfe\u3057\u3066, \\(a _ {n+1} = m\\) \u3068\u306a\u308b\u306e\u306f, \\(2^{k+1} \\leqq m' \\lt 2^{k+2}\\) \u3092\u307f\u305f\u3059\u81ea\u7136\u6570 \\(m'\\) \u306e\u3046\u3061\u306e \\(2\\) \u500b\uff08 \\(a _ n = 2m , 2m+1\\) \uff09\u3067\u3042\u308b.
        \r\n\u3053\u308c\u3092\u7e70\u8fd4\u3057\u7528\u3044\u308c\u3070, \\(a _ n = m\\) \u3068\u306a\u308b \u521d\u9805 \\(a _ 1 = N\\) \u306f, \\(2^{k+n-1} \\leqq M \\lt 2^{k+n}\\) \u3092\u307f\u305f\u3059\u81ea\u7136\u6570 \\(M\\) \u306b \\(2^{n-1}\\) \u500b\u542b\u307e\u308c\u308b.
        \r\n\u3053\u3053\u3067 \\(m = 2\\) , \\(0 \\leqq N \\lt 2^{10}\\) \u306e\u5834\u5408\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070, \u6761\u4ef6\u3092\u307f\u305f\u3059 \\(N\\) \u306f\r\n\\[\\begin{align}\r\n2 \\leqq N \\lt 2^2 \\ & \\text{\u306e\u3046\u3061} \\ 1 \\ \\text{\u500b} \\\\\r\n2^2 \\leqq N \\lt 2^3 \\ & \\text{\u306e\u3046\u3061} \\ 2 \\ \\text{\u500b} \\\\\r\n\\cdots & \\cdots \\\\\r\n2^9 \\leqq N \\lt 2^{10} \\ & \\text{\u306e\u3046\u3061} \\ 2^8 \\ \\text{\u500b} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u500b\u6570 \\(I _ 2\\) \u306f\r\n\\[\r\nI _ 2 = 1 +2 + \\cdots +2^8 = \\dfrac{2^9 -1}{2 -1} = \\underline{511} \\ .\r\n\\]\r\n

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

        (2)<\/strong> \u3068\u540c\u69d8\u306b\u8003\u3048\u308c\u3070, \\(2^k \\leqq m \\lt 2^{k+1}\\) \u3092\u307f\u305f\u3059\u81ea\u7136\u6570 \\(m\\) \u306b\u3064\u3044\u3066, \\(0 \\leqq N \\lt 2^{100}\\) \u306e\u7bc4\u56f2\u3067\r\n\\[\\begin{align}\r\nI _ m & = \\textstyle\\sum\\limits _ {n = 1}^{100-k} 2^{n-1} \\\\\r\n& = \\dfrac{2^{100-k} -1}{2 -1} = 2^{100-k} -1 \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6761\u4ef6 (\uff0a) \u3092\u307f\u305f\u3059\u306e\u306f, \\(k\\) \u304c\u6574\u6570\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{I _ m}{2^{100}} & \\leqq \\dfrac{1}{100} \\\\\r\n2^{100-k} -1 & \\leqq \\dfrac{2^{100}}{100} \\lt 2^{94} \\quad ( \\ \\text{\u2235} \\ 2^6 \\lt 100 \\lt 2^7 \\ ) \\\\\r\n\\text{\u2234} \\quad 100-k & \\lt 94 \\\\\r\n\\text{\u2234} \\quad k & \\geqq 7 \\\\ \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b \\(m\\) \u306f \\(2^7 \\leqq m \\lt 2^8\\) \u306e\u6700\u5c0f\u5024\u306a\u306e\u3067\r\n\\[\r\nm = \\underline{128} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u8ca0\u3067\u306a\u3044\u6574\u6570 \\(N\\) \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d, \\(a _ 1 = N\\) , \\(a _ {n+1} = \\left[ \\dfrac{N}{2} \\right] \\quad ( n = 1, 2, 3, \\cdots ) … \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":[119],"tags":[143,112],"class_list":["post-1150","post","type-post","status-publish","format-standard","hentry","category-nagoya_r_2014","tag-nagoya_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1150","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=1150"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1150\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1150"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1150"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}