\r\n \r\n
\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
(1)<\/strong><\/p>\r\n\u6761\u4ef6\u3088\u308a, \\(2^N \\leqq n \\leqq 2^{N-1} -1\\) \u3067\u3042\u308a, \u81ea\u7136\u6570 \\(k \\ ( 1 \\leqq k \\leqq n )\\) \u306f, \u6574\u6570 \\(\\ell \\ ( 0 \\leqq \\ell \\leqq N-1 \\ ... [1] )\\) \u3068, \u5947\u6570 \\(M \\ ( 1 \\leqq M \\leqq n \\ ... [2] )\\) \u3092\u7528\u3044\u3066\r\n\\[\r\nk = \\left\\{ \\begin{array}{ll} 2^N & ( k = 2^N \\ \\text{\u306e\u3068\u304d} ) \\\\ 2^\\ell M & ( k \\neq 2^N \\ \\text{\u306e\u3068\u304d} ) \\end{array} \\right. \\quad ... [3]\r\n\\]\r\n\u3068\u8868\u305b\u308b.
\r\n\\(P_n = 2^N A_n S_n\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nP_n = \\textstyle\\sum\\limits_{k=1}^{n} \\underline{\\dfrac{2^N A_n}{k}} _{[ \\text{A} ]}\r\n\\]\r\n\\([ \\text{A} ] = B_k\\) \u3068\u304a\u3044\u3066, \\(1 \\leqq k \\leqq n\\) \u305d\u308c\u305e\u308c\u306b\u5bfe\u3057\u3066, \\(B_k\\) \u306e\u5947\u5076\u3092\u8003\u3048\u308b\u3068<\/p>\r\n
\r\n1*<\/strong>\u3000\\(k = 2^N\\) \u306e\u3068\u304d
\r\n[3] \u3088\u308a\r\n\\[\r\nB_k = \\dfrac{2^N A_n}{2^N} = A_n\r\n\\]\r\n\u306a\u306e\u3067, \\(B_k\\) \u306f\u5947\u6570.<\/p><\/li>\r\n2*<\/strong>\u3000\\(k \\neq 2^N\\) \u306e\u3068\u304d
\r\n[3] \u3088\u308a\r\n\\[\r\nB_k = \\dfrac{2^N A_n}{2^{\\ell} M} = 2^{N -\\ell} \\cdot \\dfrac{A_n}{M}\r\n\\]\r\n[1] \u3088\u308a, \\(2^{N -\\ell}\\) \u306f\u5076\u6570, [2] \u3088\u308a \\(\\dfrac{A_n}{M}\\) \u306f\u5947\u6570\u306a\u306e\u3067, \\(B_k\\) \u306f\u5076\u6570.<\/p><\/li>\r\n<\/ol>\r\n\u4ee5\u4e0a\u3088\u308a, \\(P_n\\) \u306f \\(1\\) \u3064\u306e\u5947\u6570\u3068 \\(n-1\\) \u500b\u306e\u5076\u6570\u306e\u548c\u3067\u3042\u308a, \u3088\u3063\u3066\u5947\u6570\u3067\u3042\u308b.<\/p>\r\n
(2)<\/strong><\/p>\r\n\\[\r\nS_n = \\dfrac{P_n}{2^N A_n}\r\n\\]\r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(P_n , A_n\\) \u306f\u3068\u3082\u306b\u5947\u6570\u306a\u306e\u3067, \\(S_n\\) \u306e\u5206\u6bcd\u306f, \\(2^{N+1}\\) \u3067\u5272\u308a\u5207\u308c\u308b\u3053\u3068\u306f\u306a\u3044.
\r\n\u6761\u4ef6\u3088\u308a, \\(S_n = \\dfrac{40+m}{20}\\) \u3067, \\(20 = 2^2 \\cdot 5\\) \u306a\u306e\u3067\r\n\\[\r\nN \\leqq 2\r\n\\]\r\n\u3086\u3048\u306b, \\(n\\) \u306e\u5019\u88dc\u306f, \\(1 \\leqq n \\leqq 2^3-1 = 7\\) .
\r\n\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066, \\(S_n\\) \u3092\u6c42\u3081\u308b\u3068,\r\n\\[\\begin{align}\r\n\\begin{array}{c|ccccc} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\\\ \\hline S_n & 1 & \\dfrac{3}{2} & \\dfrac{11}{6} & \\dfrac{25}{12} & \\dfrac{137}{60} & \\dfrac{49}{20} & \\dfrac{363}{140} \\end{array}\r\n\\end{align}\\]\r\n\\(m \\gt 0\\) \u3088\u308a, \\(S_n \\gt 2\\) \u306a\u306e\u3067, \\(n \\geqq 4\\) .
\r\n\u3053\u306e\u3046\u3061, \\(n=6\\) \u306e\u3068\u304d\u306e\u307f, \u5206\u6bcd\u304c \\(20\\) \u306e\u7d04\u6570\u3068\u306a\u308b\u306e\u3067, \u6c42\u3081\u308b\u6574\u6570\u306e\u7d44\u306f\r\n\\[\r\n(m,n) = \\underline{( 6 , 9 )}\r\n\\]\r\n(3)<\/strong><\/p>\r\n\u5b9f\u6570 \\(x\\) \u306e\u5c0f\u6570\u90e8\u5206\u3092 \\(\\langle x \\rangle\\) \u3068\u304a\u304f.
\r\n(1)<\/strong> \u306e\u7d50\u679c\u304b\u3089, \u5947\u6570 \\(r \\ ( 1 \\leqq r \\leqq 15 )\\) \u3092\u7528\u3044\u3066\r\n\\[\r\nb = \\left\\langle A _ {20} S _ {20} \\right\\rangle = \\dfrac{r}{16}\r\n\\]\r\n\u3068\u8868\u305b\u308b.
\r\n\\[\\begin{align}\r\nA _ {20} S _ {20} & = \\underline{A _ {20} \\left( 1 +\\dfrac{1}{3} +\\dfrac{1}{5} +\\cdots +\\dfrac{1}{19} \\right)} _ {[1]} \\\\\r\n& \\quad +\\underline{A _ {20} \\left( \\dfrac{1}{2} +\\dfrac{1}{6} +\\dfrac{1}{10} +\\dfrac{1}{14} +\\dfrac{1}{18} \\right)} _ {[2]} \\\\\r\n& \\qquad +A _ {20} \\underline{\\left( \\dfrac{1}{4} +\\dfrac{1}{12} +\\dfrac{1}{20} \\right)} _ {[3]} \\\\\r\n& \\qquad \\quad +\\underline{A _ {20} \\cdot \\dfrac{1}{8}} _ {[4]} +\\underline{A _ {20} \\cdot \\dfrac{1}{16}} _ {[5]} \\ .\r\n\\end{align}\\]\r\n\\(\\dfrac{A_n}{M}\\) \u306f\u5947\u6570\uff08\u6574\u6570\uff09\u3068\u306a\u308b\u3053\u3068\u306b\u7740\u76ee\u3059\u308c\u3070, [1] \u306f\u6574\u6570, \u3064\u307e\u308a\r\n\\[\r\n\\langle [1] \\rangle = 0 \\quad ... [6] \\ .\r\n\\]\r\n\u307e\u305f, \u5947\u6570 \\(5\\) \u500b\u306e\u548c\u306f\u5947\u6570\u3068\u306a\u308b\u306e\u3067,\r\n\\[\\begin{align}\r\n\\langle [2] \\rangle & = \\left\\langle \\dfrac{1}{2} \\left( A _ {20} +\\dfrac{A _ {20}}{3} +\\dfrac{A _ {20}}{5} +\\dfrac{A _ {20}}{7} +\\dfrac{A _ {20}}{9} \\right) \\right\\rangle \\\\\r\n& = \\dfrac{1}{2} \\ .\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \u6cd5 \\(16\\) \u306e\u5408\u540c\u5f0f\u3092\u8003\u3048\u308b\u3068\r\n\\[\\begin{align}\r\nA _ {20} & \\equiv 1 \\cdot 3 \\cdot 5 \\cdot 7 \\cdot (-7) \\cdot (-5) \\cdot (-3) \\cdot (-1)\\cdot 1 \\cdot 3 \\\\\r\n& \\equiv 15^2 \\cdot 49 \\cdot 3 \\\\\r\n& \\equiv (-1)^2 \\cdot 1 \\cdot 3 \\\\\r\n& \\equiv 3 \\quad ( \\text{mod} 16 ) \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nA _ {20} \\equiv 3 \\quad ( \\text{mod} 8 ) , \\quad A _ {20} \\equiv 3 \\quad ( \\text{mod} 4 )\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\langle [4] \\rangle = \\dfrac{3}{8} , \\quad \\langle [5] \\rangle = \\dfrac{3}{16} \\ .\r\n\\]\r\n\u307e\u305f, [3] \u306b\u3064\u3044\u3066\u306f\r\n\\[\r\n[3] = \\dfrac{1}{4} A _ {20} \\cdot \\dfrac{23}{15}\r\n\\]\r\n\u3067\u3042\u308a, \u6cd5 \\(4\\) \u306e\u5408\u540c\u5f0f\u3092\u8003\u3048\u308b\u3068\r\n\\[\r\n23 \\equiv 15 \\equiv 3 \\quad ( \\text{mod} 4 )\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\langle [3] \\rangle = \\dfrac{1}{4} \\cdot 3 \\cdot \\dfrac{3}{3} = \\dfrac{3}{4}\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a\r\n\\[\\begin{align}\r\nb & = \\left\\langle 0 +\\dfrac{1}{2} +\\dfrac{3}{4} +\\dfrac{3}{8} +\\dfrac{3}{16} \\right\\rangle \\\\\r\n& = \\left\\langle \\dfrac{8 +12 +6 +3}{16} \\right\\rangle = \\underline{\\dfrac{13}{16}}\r\n\\end{align}\\]\r\n\r\n \r\n « \u89e3\u7b54\u3092\u96a0\u3059 <\/a>\r\n <\/p>\r\n <\/div>","protected":false},"excerpt":{"rendered":"\u6b63\u306e\u6574\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066 \\[ S _ n = \\textstyle\\sum\\limits _ {k=1}^{n} \\dfrac{1}{k} \\] \u3068\u304a\u304d, \\(1\\) \u4ee5\u4e0a \\(n\\) \u4ee5\u4e0b\u306e\u3059\u3079\u3066\u306e\u5947\u6570\u306e\u7a4d\u3092 […]","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":[152],"tags":[142,162],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1442"}],"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=1442"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1442\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1442"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1442"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}