{"id":211,"date":"2025-10-19T16:27:03","date_gmt":"2025-10-19T16:27:03","guid":{"rendered":"https:\/\/www.starluo.top\/?p=211"},"modified":"2025-10-19T16:27:04","modified_gmt":"2025-10-19T16:27:04","slug":"e4-j%e5%bf%ab%e9%80%9f%e6%b1%82%e5%ba%8f%e5%88%97%e4%b8%ad%e4%ba%92%e8%b4%a8%e7%9a%84%e6%95%b4%e6%95%b0%e5%af%b9%e7%9a%84%e4%b8%aa%e6%95%b0","status":"publish","type":"post","link":"https:\/\/www.starluo.top\/?p=211","title":{"rendered":"[E4-J]\u5feb\u901f\u6c42\u5e8f\u5217\u4e2d\u4e92\u8d28\u7684\u6574\u6570\u5bf9\u7684\u4e2a\u6570"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">\u9898\u76ee<\/h2>\n\n\n\n<p>\u7ed9\u51fa$n$\u4e2a\u6570\u00a0$a_1,a_2,\\cdots,a_n$\uff0c\u6c42\u51fa\u6709\u591a\u5c11\u4e2a\u6574\u6570\u5bf9$(i,j)$\uff0c\u6ee1\u8db3$1\u2264i&lt;j\u2264n$\uff0c\u4e14$a_i$\u4e0e$a_j$\u4e92\u8d28\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u5206\u6790<\/h2>\n\n\n\n<p>Deepseek\u5982\u662f\u8bf4\uff1a<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u76f4\u63a5\u65b9\u6cd5\u7684\u5c40\u9650\u6027<\/h3>\n\n\n\n<p>\u6700\u76f4\u63a5\u7684\u65b9\u6cd5\u662f\u68c0\u67e5\u6240\u6709\u53ef\u80fd\u7684\u6570\u5bf9\uff1a<\/p>\n\n\n\n<pre class=\"wp-block-preformatted\">count = 0\nfor i in range(n):\n    for j in range(i+1, n):\n        if gcd(arr[i], arr[j]) == 1:\n            count += 1<\/pre>\n\n\n\n<p>\u8fd9\u79cd\u65b9\u6cd5\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u662fO(n\u00b2)\uff0c\u5f53n\u5f88\u5927\u65f6\u6548\u7387\u5f88\u4f4e\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5bb9\u65a5\u539f\u7406\u7684\u601d\u8def<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">\u57fa\u672c\u601d\u60f3<\/h4>\n\n\n\n<p>\u6211\u4eec\u53ef\u4ee5\u6362\u4e2a\u89d2\u5ea6\u601d\u8003\uff1a<strong>\u6240\u6709\u6570\u5bf9\u7684\u6570\u91cf - \u4e0d\u4e92\u8d28\u7684\u6570\u5bf9\u6570\u91cf<\/strong><\/p>\n\n\n\n<p>\u4f46\u662f\"\u4e0d\u4e92\u8d28\"\u7684\u60c5\u51b5\u5f88\u590d\u6742\uff0c\u56e0\u4e3a\u4e24\u4e2a\u6570\u53ef\u80fd\u6709\u591a\u4e2a\u516c\u56e0\u6570\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u5173\u952e\u6d1e\u5bdf<\/h4>\n\n\n\n<p>\u4e24\u4e2a\u6570\u4e92\u8d28 \u21d4 \u5b83\u4eec\u6ca1\u6709\u5171\u540c\u7684\u8d28\u56e0\u6570<\/p>\n\n\n\n<p>\u6211\u4eec\u53ef\u4ee5\u7528<strong>\u5bb9\u65a5\u539f\u7406<\/strong>\u6765\u8ba1\u6570\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5148\u8ba1\u7b97\u6240\u6709\u6570\u5bf9<\/li>\n\n\n\n<li>\u51cf\u53bb\u81f3\u5c11\u67091\u4e2a\u516c\u5171\u8d28\u56e0\u6570\u7684\u6570\u5bf9<\/li>\n\n\n\n<li>\u52a0\u56de\u81f3\u5c11\u67092\u4e2a\u516c\u5171\u8d28\u56e0\u6570\u7684\u6570\u5bf9\uff08\u56e0\u4e3a\u88ab\u51cf\u4e86\u4e24\u6b21\uff09<\/li>\n\n\n\n<li>\u51cf\u53bb\u81f3\u5c11\u67093\u4e2a\u516c\u5171\u8d28\u56e0\u6570\u7684\u6570\u5bf9<\/li>\n\n\n\n<li>...<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\u7684\u4f5c\u7528<\/h3>\n\n\n\n<p>\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\u03bc(d)\u6b63\u597d\u63d0\u4f9b\u4e86\u6211\u4eec\u9700\u8981\u7684\u7b26\u53f7\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03bc(1) = 1<\/li>\n\n\n\n<li>\u5982\u679cd\u6709\u91cd\u590d\u8d28\u56e0\u6570\uff0c\u03bc(d) = 0\uff08\u4e0d\u8003\u8651\u8fd9\u79cd\u60c5\u51b5\uff09<\/li>\n\n\n\n<li>\u5982\u679cd\u6709k\u4e2a\u4e0d\u540c\u8d28\u56e0\u6570\uff0c\u03bc(d) = (-1)^k<\/li>\n<\/ul>\n\n\n\n<p>\u8fd9\u6b63\u597d\u5bf9\u5e94\u5bb9\u65a5\u539f\u7406\u4e2d\u5404\u9879\u7684\u7b26\u53f7\uff01<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6570\u5b66\u63a8\u5bfc<\/h3>\n\n\n\n<h4 class=\"wp-block-heading\">\u516c\u5f0f<\/h4>\n\n\n\n<p>\u4e92\u8d28\u5bf9\u7684\u6570\u91cf = \u2211 \u03bc(d) \u00d7 C(c_d, 2)<\/p>\n\n\n\n<p>\u5176\u4e2d\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>d\u904d\u5386\u6240\u6709\u53ef\u80fd\u7684\u516c\u7ea6\u6570\uff081\u5230M\uff09<\/li>\n\n\n\n<li>c_d = \u80fd\u88abd\u6574\u9664\u7684\u6570\u7684\u4e2a\u6570<\/li>\n\n\n\n<li>C(c_d, 2) = \u4ecec_d\u4e2a\u6570\u4e2d\u90092\u4e2a\u7684\u7ec4\u5408\u6570<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">\u4e3a\u4ec0\u4e48\u8fd9\u4e2a\u516c\u5f0f\u6709\u6548\uff1f<\/h4>\n\n\n\n<p>\u5bf9\u4e8e\u4efb\u610f\u4e24\u4e2a\u6570a\u548cb\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679cgcd(a,b) = 1\uff0c\u90a3\u4e48\u53ea\u6709d=1\u65f6\u4f1a\u88ab\u8ba1\u6570\uff0c\u8d21\u732e\u4e3a\u03bc(1)\u00d71 = 1<\/li>\n\n\n\n<li>\u5982\u679cgcd(a,b) = g > 1\uff0c\u90a3\u4e48\u5bf9\u4e8eg\u7684\u6bcf\u4e2a\u56e0\u6570d\uff0c\u8fd9\u4e2a\u6570\u5bf9\u90fd\u4f1a\u88ab\u7edf\u8ba1<\/li>\n<\/ul>\n\n\n\n<p>\u901a\u8fc7\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\u7684\u6027\u8d28\uff0c\u6240\u6709\u975e\u4e92\u8d28\u5bf9\u7684\u8d21\u732e\u4f1a\u76f8\u4e92\u62b5\u6d88\uff0c\u53ea\u5269\u4e0b\u4e92\u8d28\u5bf9\u7684\u8d21\u732e\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u5177\u4f53\u4f8b\u5b50\u6f14\u793a<\/h3>\n\n\n\n<p>\u5047\u8bbe\u6570\u7ec4\u4e3a[2, 3, 4, 6]\uff0c\u6700\u5927\u6570M=6\u3002<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">\u6b65\u9aa41\uff1a\u8ba1\u7b97c_d<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>d=1\uff1a\u6240\u6709\u6570\u90fd\u80fd\u88ab1\u6574\u9664\uff0cc\u2081=4<\/li>\n\n\n\n<li>d=2\uff1a\u80fd\u88ab2\u6574\u9664\u7684\u6570\u6709[2,4,6]\uff0cc\u2082=3<\/li>\n\n\n\n<li>d=3\uff1a\u80fd\u88ab3\u6574\u9664\u7684\u6570\u6709[3,6]\uff0cc\u2083=2<\/li>\n\n\n\n<li>d=4\uff1a\u80fd\u88ab4\u6574\u9664\u7684\u6570\u6709[4]\uff0cc\u2084=1<\/li>\n\n\n\n<li>d=5\uff1a\u6ca1\u6709\u6570\u80fd\u88ab5\u6574\u9664\uff0cc\u2085=0<\/li>\n\n\n\n<li>d=6\uff1a\u80fd\u88ab6\u6574\u9664\u7684\u6570\u6709[6]\uff0cc\u2086=1<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">\u6b65\u9aa42\uff1a\u8ba1\u7b97\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570<\/h4>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u03bc(1) = 1<\/li>\n\n\n\n<li>\u03bc(2) = -1\uff081\u4e2a\u8d28\u56e0\u6570\uff09<\/li>\n\n\n\n<li>\u03bc(3) = -1\uff081\u4e2a\u8d28\u56e0\u6570\uff09<\/li>\n\n\n\n<li>\u03bc(4) = 0\uff08\u6709\u5e73\u65b9\u56e0\u5b502\u00b2\uff09<\/li>\n\n\n\n<li>\u03bc(5) = -1\uff081\u4e2a\u8d28\u56e0\u6570\uff09<\/li>\n\n\n\n<li>\u03bc(6) = 1\uff082\u4e2a\u4e0d\u540c\u8d28\u56e0\u65702\u548c3\uff09<\/li>\n<\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">\u6b65\u9aa43\uff1a\u5e94\u7528\u516c\u5f0f<\/h4>\n\n\n\n<pre class=\"wp-block-preformatted\">\u603b\u4e92\u8d28\u5bf9\u6570 = \u03bc(1)\u00d7C(4,2) + \u03bc(2)\u00d7C(3,2) + \u03bc(3)\u00d7C(2,2) + \u03bc(4)\u00d7C(1,2) + \u03bc(5)\u00d7C(0,2) + \u03bc(6)\u00d7C(1,2)\n          = 1\u00d76 + (-1)\u00d73 + (-1)\u00d71 + 0\u00d70 + (-1)\u00d70 + 1\u00d70\n          = 6 - 3 - 1 = 2<\/pre>\n\n\n\n<h4 class=\"wp-block-heading\">\u9a8c\u8bc1<\/h4>\n\n\n\n<p>\u6570\u7ec4[2,3,4,6]\u7684\u4e92\u8d28\u5bf9\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>(2,3)\uff1agcd=1 \u2713<\/li>\n\n\n\n<li>(2,4)\uff1agcd=2 \u2717<\/li>\n\n\n\n<li>(2,6)\uff1agcd=2 \u2717<\/li>\n\n\n\n<li>(3,4)\uff1agcd=1 \u2713<\/li>\n\n\n\n<li>(3,6)\uff1agcd=3 \u2717<\/li>\n\n\n\n<li>(4,6)\uff1agcd=2 \u2717<\/li>\n<\/ul>\n\n\n\n<p>\u786e\u5b9e\u67092\u4e2a\u4e92\u8d28\u5bf9\uff0c\u4e0e\u8ba1\u7b97\u7ed3\u679c\u4e00\u81f4\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u7b97\u6cd5\u4f18\u52bf<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u65f6\u95f4\u590d\u6742\u5ea6\uff1aO(M log M)\uff0c\u5176\u4e2dM\u662f\u6700\u5927\u6570<\/li>\n\n\n\n<li>\u5f53n\u5f88\u5927\u4f46M\u76f8\u5bf9\u8f83\u5c0f\u65f6\uff0c\u6bd4O(n\u00b2)\u65b9\u6cd5\u5feb\u5f88\u591a<\/li>\n\n\n\n<li>\u5229\u7528\u4e86\u6570\u8bba\u6027\u8d28\uff0c\u907f\u514d\u4e86\u91cd\u590d\u8ba1\u7b97<\/li>\n<\/ul>\n\n\n\n<p>\u8fd9\u5c31\u662f\u4e3a\u4ec0\u4e48\u8fd9\u4e2a\u57fa\u4e8e\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\u548c\u5bb9\u65a5\u539f\u7406\u7684\u7b97\u6cd5\u80fd\u591f\u9ad8\u6548\u89e3\u51b3\u4e92\u8d28\u5bf9\u8ba1\u6570\u95ee\u9898\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u9898\u89e3\uff08Possible\uff09<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;stdio.h>\n#include&lt;math.h>\nint i,j,n,a&#91;1000001],num&#91;1000001]={1,1},p&#91;1000001]={0},flag,pc=0,d&#91;1000001]={0},mx=0,cnt&#91;1000001]={0},t,mu,ans=0;\nint main()\n{\n\tfor(i=2;i&lt;1000001;i++)\n\t{\n\t\tif(num&#91;i]) continue;\n\t\tnum&#91;i]=1,flag=0;\n\t\tfor(j=2;j&lt;sqrt(i);j++)\n\t\t{\n\t\t\tif(!(i%j))\n\t\t\t{\n\t\t\t\tflag=1;\n\t\t\t\tbreak;\n\t\t\t}\n\t\t}\n\t\tif(flag) continue;\n\t\tnum&#91;i]=0;\n\t\tp&#91;pc++]=i;\n\t\tfor(j=2;;j++)\n\t\t{\n\t\t\tif(i*j>1000000) break;\n\t\t\tnum&#91;i*j]=1;\n\t\t}\n\t}\n\tscanf(\"%d\",&amp;n);\n\tfor(i=0;i&lt;n;i++)\n\t{\n\t\tscanf(\"%d\",&amp;a&#91;i]);\n\t\tmx=mx>a&#91;i]?mx:a&#91;i];\n\t\tfor(j=1;j&lt;=a&#91;i];j++) if(!(a&#91;i]%j)) d&#91;j]++;\n\t}\n\tfor(i=1;i&lt;=mx;i++)\n\t{\n\t\tt=i,mu=1;\n\t\tfor(j=0;j&lt;pc;j++)\n\t\t{\n\t\t\tif(!num&#91;i])\n\t\t\t{\n\t\t\t\tcnt&#91;i]++;\n\t\t\t\tbreak;\n\t\t\t}\n\t\t\tif(!(t%p&#91;j])) t\/=p&#91;j],cnt&#91;i]++;\n\t\t\tif(!(t%p&#91;j]))\n\t\t\t{\n\t\t\t\tmu=0;\n\t\t\t\tbreak;\n\t\t\t}\n\t\t\tif(t==1||t&lt;p&#91;j]) break;\n\t\t}\n\t\tif(mu) mu=cnt&#91;i]%2?-1:1;\n\t\tans+=mu*d&#91;i]*(d&#91;i]-1)\/2;\n\t\t\/\/printf(\"mu=%d d&#91;%d]=%d\\n\",mu,i,d&#91;i]); \n\t}\n\tprintf(\"%d\",ans);\n\treturn 0;\n}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>\u9898\u76ee \u7ed9\u51fa$n$\u4e2a\u6570\u00a0$a_1,a_2,\\cdots,a_n$\uff0c\u6c42\u51fa\u6709\u591a\u5c11\u4e2a\u6574\u6570\u5bf9$(i,j)$\uff0c\u6ee1\u8db3$1\u2264i&lt;j\u2264n$\uff0c\u4e14 &#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"emotion":"","emotion_color":"","title_style":"","license":"","footnotes":""},"categories":[8],"tags":[],"class_list":["post-211","post","type-post","status-publish","format-standard","hentry","category-oi"],"_links":{"self":[{"href":"https:\/\/www.starluo.top\/index.php?rest_route=\/wp\/v2\/posts\/211","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.starluo.top\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.starluo.top\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.starluo.top\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.starluo.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=211"}],"version-history":[{"count":1,"href":"https:\/\/www.starluo.top\/index.php?rest_route=\/wp\/v2\/posts\/211\/revisions"}],"predecessor-version":[{"id":212,"href":"https:\/\/www.starluo.top\/index.php?rest_route=\/wp\/v2\/posts\/211\/revisions\/212"}],"wp:attachment":[{"href":"https:\/\/www.starluo.top\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=211"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.starluo.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=211"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.starluo.top\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}