{"id":375,"date":"2022-09-20T18:22:19","date_gmt":"2022-09-20T10:22:19","guid":{"rendered":"https:\/\/blog.cirno.fun\/?p=375"},"modified":"2024-09-12T09:33:46","modified_gmt":"2024-09-12T01:33:46","slug":"%e7%ba%bf%e6%80%a7%e7%ad%9b%e6%b1%82%e6%ac%a7%e6%8b%89%e5%87%bd%e6%95%b0","status":"publish","type":"post","link":"https:\/\/blog.sssn.tech\/?p=375","title":{"rendered":"\u7ebf\u6027\u7b5b\u6c42\u6b27\u62c9\u51fd\u6570"},"content":{"rendered":"

\u4e0a\u8282\u8bf4\u9053\uff0c\u6c42\u4e00\u4e2a\u6570\u7684\u6b27\u62c9\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7b49\u540c\u4e8e\u5bf9\u5176\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff0c\u90fd\u662f<\/p>\n

O(\\sqrt N)<\/code><\/pre>\n
\u5982\u679c\u9898\u76ee\u8981\u6c42\u6c42\u67d0\u4e2a\u533a\u95f4\u4e0a\u6240\u6709\u6570\u7684\u6b27\u62c9\u51fd\u6570\uff0c\u518d\u7528\u8fd9\u4e2a\u65b9\u6cd5\u7684\u8bdd\uff0c\u65f6\u95f4\u590d\u6742\u5ea6\u5c31\u662f<\/p>\n
O(N\\sqrt N)<\/code><\/pre>\n
\u672c\u8282\u4ecb\u7ecd\u7528\u7ebf\u6027\u7b5b\u6cd5\u6c42\u6b27\u62c9\u51fd\u6570\uff0c\u4ed6\u540c\u6837\u53ef\u4ee5\u4ee5\u7ebf\u6027(O(N))\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u89e3\u51b3\u4e0a\u8ff0\u95ee\u9898 \u5728\u6b64\u4e4b\u524d\uff0c\u5148\u590d\u4e60\u7d20\u6570\u7684\u7ebf\u6027\u7b5b\u6cd5\uff1a<\/p>\n
void get_primes(int n){\/\/\u6c42n\u4ee5\u5185\u7684\u6240\u6709\u7d20\u6570\n    for(int i=2;i<=n\/i;i++){\/\/sqrt,i*i\u7684\u5199\u6cd5\u90fd\u4e0d\u5982\u8fd9\u4e2a\u597d\n        if(!v[i])   primes[++cnt]=i;\n        for(int j=1;primes[j]<=n\/i;j++){\n            v[primes[j]*i]=true;\n            if(i%primes[j]==0)  break;\/\/\u56e0\u4e3a\u8fd9\u884c\uff0c\u7ebf\u6027\u7b5b\u624d\u662f\u7ebf\u6027\u590d\u6742\u5ea6\n        }\n    }\n}<\/code><\/pre>\n
\u5728\u5145\u5206\u7406\u89e3\u7ebf\u6027\u7b5b\u7684\u6761\u4ef6\u4e0b\uff0c\u7ed9\u51fa\u6b27\u62c9\u51fd\u6570\u7684\u91cd\u8981\u6027\u8d28\uff1a<\/p>\n
\\phi(ab)=\\frac{\\phi(a) \\phi(b) gcd(a,b)}{\\phi(gcd(a,b))} <\/code><\/pre>\n
\u8fd9\u662f\u7531\u4e8e\u6b27\u62c9\u51fd\u6570\u662f\u79ef\u6027\u51fd\u6570\u800c\u5f97\u6765\u7684\uff0c\u5177\u4f53\u8bc1\u660e\u4ece\u7565\uff0c\u5982\u679c\u6709\u673a\u4f1a\u4e13\u95e8\u5199\u4e00\u7bc7 \u5728\u8fd9\u4e2a\u5f0f\u5b50\u4e2d\uff0c\u5305\u542b\u4e86\u4efb\u610fab\u60c5\u51b5\uff0c\u6211\u4eec\u73b0\u5728\u89c2\u5bdf\u7ebf\u6027\u7b5b\u6cd5\u7684\u5185\u5bb9\uff1a<\/p>\n
    \n
  1. \u5982\u679ci\u662f\u8d28\u6570\uff0c\u663e\u7136phi(i)=i-1,\u8fd9\u662f\u56e0\u4e3a\u9664\u4e86\u4ed6\u81ea\u5df1\uff0c\u4efb\u4f55\u6bd4\u4ed6\u5c0f\u7684\u6b63\u6574\u6570\u90fd\u548c\u4ed6\u4e92\u8d28<\/li>\n
  2. \u5982\u679ci\u662f\u5408\u6570\uff0c\u5728\u7ebf\u6027\u7b5b\u6cd5\u4e2d\uff0c\u4ed6\u4f1a\u88ab\u7b5b\u53bb\uff0c\u800c\u88ab\u7b5b\u7684\u60c5\u51b5\u53ef\u4ee5\u5206\u4e3a\u4e24\u79cd<\/li>\n<\/ol>\n
    \u5bf9\u4e8e\u6211\u4eec\u5c06\u8981\u7b5b\u53bb\u7684\u6570primes[j]*i:<\/p>\n
      \n
    1. \u5f53i%primes[j]==0\u65f6\uff1a i\u662fprimes[j]\u7684\u6574\u6570\u500d\u6570\uff0c\u6240\u4ee5\u663e\u7136gcd(i,primes[j])=primes[j]<\/code>\u6240\u4ee5 “`katex
      \n\\phi(i\\cdot primes[j])=\\frac{\\phi(i) \\phi(primes[j]) gcd(i,primes[j])}{\\phi(gcd(i,primes[j]))}  \\
      \n=\\phi(i)\\cdot primes[j]<\/p>\n
      <\/code><\/pre>\n<\/li>\n
    2. \n
      \u5f53i%primes[j]!=0\u65f6\uff1a \u6b64\u65f6i\u548cprimes[j]\u662f\u4e92\u8d28\u7684\uff0c\u6240\u4ee5\u663e\u7136gcd(i,primes[j]=1)<\/code>\u53c8\u56e0\u4e3aprimes[j]\u662f\u8d28\u6570\uff0c\u6240\u4ee5 “`katex
      \n\\phi(i\\cdot primes[j])=\\phi(i)\\cdot \\phi(primes[j])=\\phi(i) \\cdot (primes[j]-1)<\/p>\n
      \n\n\u81f3\u6b64\uff0c\u6211\u4eec\u5c31\u89e3\u51b3\u4e86\u7528\u7ebf\u6027\u7b5b\u8d28\u6570\u6c42\u6b27\u62c9\u51fd\u6570\u7684\u95ee\u9898<\/code><\/pre>\n<\/li>\n<\/ol>\n
      \u4e0a\u6e90\u7801\u4e86<\/p>\n
      int primes[N],cnt,phi[N];\nbool v[N];\nvoid get_phi(int n){\n    phi[1]=1;\/\/\u89c4\u5b9a\n    for(int i=2;i<=n;i++){\n        if(!v[i]){\n            primes[++cnt]=i;\n            phi[i]=i-1;\n        }\n        for(int j=1;primes[j]<=n\/i;j++){\n            v[primes[j]*i]=true;\n            if(i%primes[j]==0){\n                phi[i*primes[j]]=phi[i]*primes[j];\n                break;\n            }\n            phi[i*primes[j]]=phi[i]*(primes[j]-1);\n        }\n    }\n}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"
      \u4e0a\u8282\u8bf4\u9053\uff0c\u6c42\u4e00\u4e2a\u6570\u7684\u6b27\u62c9\u51fd\u6570\u7684\u65f6\u95f4\u590d\u6742\u5ea6\u7b49\u540c\u4e8e\u5bf9\u5176\u8fdb\u884c\u56e0\u5f0f\u5206\u89e3\uff0c\u90fd\u662f O(\\sq […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[13,16],"tags":[],"class_list":["post-375","post","type-post","status-publish","format-standard","hentry","category-13","category-16"],"_links":{"self":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/375","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=375"}],"version-history":[{"count":1,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/375\/revisions"}],"predecessor-version":[{"id":748,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=\/wp\/v2\/posts\/375\/revisions\/748"}],"wp:attachment":[{"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=375"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=375"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.sssn.tech\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=375"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}