{"id":805,"date":"2020-09-16T16:08:19","date_gmt":"2020-09-16T08:08:19","guid":{"rendered":"http:\/\/www.zyhcoding.club\/?p=805"},"modified":"2021-05-07T18:24:12","modified_gmt":"2021-05-07T10:24:12","slug":"%e8%8e%ab%e6%af%94%e4%b9%8c%e6%96%af%e5%8f%8d%e6%bc%94","status":"publish","type":"post","link":"http:\/\/www.zyhcoding.club\/index.php\/2020\/09\/16\/%e8%8e%ab%e6%af%94%e4%b9%8c%e6%96%af%e5%8f%8d%e6%bc%94\/","title":{"rendered":"\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14"},"content":{"rendered":"

\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\uff1ad=1,\u03bc(d)=1; d=\\prod_{i=1}^{k}{p}_{i}, \u5373d\u7684\u8d28\u56e0\u5b50\u4e2d\u6ca1\u6709\u91cd\u590d\u7684,\u03bc(d)=(-1)^k<\/code>; \u5426\u5219\u03bc(d)=0<\/code><\/p>\n

\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\u7684\u7ebf\u6027\u6c42\u6cd5\uff1a<\/p>\n

void Mobius(){\n    mu[1] = 1, cnt = 0, vis[1] = 1;\n    for (ll i=2; i<=n; i++){\n        if (!vis[i])    prime[++cnt] = i, mu[i] = -1;\n        for (int j=1; j<=cnt && i*prime[j]<=n; j++){\n            vis[i*prime[j]] = 1;\n            if (i%prime[j]==0)  break;\n            else   mu[i*prime[j]] = -mu[i];\n        }\n    }\n    for (int i=1; i<=n; i++)    mu[i] += mu[i-1];\n}<\/code><\/pre>\n
\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14<\/h2>\n
\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14\u662f\u6570\u8bba\u4e2d\u91cd\u8981\u4e14\u6bd4\u8f83\u96be\u7684\u4e00\u90e8\u5206\uff0c\u5bf9\u4e8e\u4e00\u4e9b\u51fd\u6570f(n)<\/code>\uff0c\u5f53\u5f88\u96be\u76f4\u63a5\u6c42\u51fa\u5b83\u7684\u503c\uff0c\u800c\u5f88\u5bb9\u6613\u6c42\u51fa\u5b83\u7684\u7ea6\u6570\u6216\u8005\u500d\u6570\u548c\u65f6\uff0c\u53ef\u4ee5\u901a\u8fc7\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14\u7b80\u5316\u8fd0\u7b97\uff0c\u6c42\u5f97f(n)<\/code>\u3002(\u5176\u5b9e\u8fd9\u5757\u4e3b\u8981\u5c31\u662f\u63a8\u5f0f\u5b50...)<\/p>\n
\u672c\u6587\u7ae0\u8ba8\u8bba\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14\u7684\u6240\u6709\u524d\u7f6e\u5185\u5bb9\u4ee5\u53ca\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14\u7684\u4e00\u822c\u5957\u8def\uff0c\u8fd8\u6709\u76f8\u5173\u7684\u4e00\u4e9b\u4f8b\u9898\u3002<\/p>\n
\n
\u51e0\u4e2a\u5e38\u7528\u5230\u7684\u516c\u5f0f<\/h4>\n
\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570<\/strong>
\nd = 1\u65f6\uff0c\u03bc(d)=1\uff1b
\nd\u4e3ak\u4e2a\u53ea\u51fa\u73b01\u6b21\u7684\u8d28\u6570\u76f8\u4e58\u65f6\uff0c\u03bc(d)=(-1)^k<\/code>\uff1b
\n\u5bf9\u4e8e\u5176\u5b83\u60c5\u51b5\uff0c\u03bc(d) = 0.
\n\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\u662f\u79ef\u6027\u51fd\u6570\u3002<\/p>\n
\u79ef\u6027\u51fd\u6570<\/h4>\n
\u82e5\u51fd\u6570f(x)<\/code>\u6ee1\u8db3\u5f53gcd(a, b)=1<\/code>\u65f6\u6709f(ab) = f(a)f(b)<\/code>\uff0c\u5219f(x)<\/code>\u4e3a\u79ef\u6027\u51fd\u6570\u3002
\n\u82e5\u51fd\u6570f(x)<\/code>\u6ee1\u8db3\u5bf9\u4efb\u610f\u6b63\u6574\u6570a, b\u6709f(ab) = f(a)f(b)<\/code>\uff0c\u5219f(x)<\/code>\u4e3a\u5b8c\u5168\u79ef\u6027\u51fd\u6570\u3002<\/p>\n
\u4efb\u4f55\u4e00\u4e2a\u79ef\u6027\u51fd\u6570\u90fd\u53ef\u4ee5\u7ebf\u6027\u7b5b\u51fa\u6765\uff0c\u5c31\u50cf\u6b27\u62c9\u51fd\u6570\uff0c\u83ab\u6bd4\u4e4c\u65af\u51fd\u6570\u4e00\u6837\u3002<\/p>\n
\u5355\u4f4d\u51fd\u6570<\/strong>\uff0c\u4e5f\u53eb\u72c4\u5229\u514b\u96f7\u5355\u4f4d\u5143\uff0c\\varepsilon(x) = [x=1]<\/code>\uff0cemm\u5728\u5e73\u9762\u4e0a\u5c31\u662f\u4e00\u4e2a\u70b9\u3002(\u5b8c\u5168\u79ef\u6027\u51fd\u6570)<\/p>\n
\u5e38\u6570\u51fd\u6570<\/strong>\uff0cI(x) = 1<\/code>\u3002 (\u5b8c\u5168\u79ef\u6027\u51fd\u6570)<\/p>\n
\u6052\u7b49\u51fd\u6570<\/strong>\uff0cid(x) = x<\/code>\u3002(\u5b8c\u5168\u79ef\u6027\u51fd\u6570)<\/p>\n
\u6b27\u62c9\u51fd\u6570<\/strong>
\nx\u7684\u6b27\u62c9\u51fd\u6570\\phi(x)<\/code>\u7684\u503c\u4e3a\u5728[1, x-1]\u4e2d\uff0c\u6709\u591a\u5c11\u4e2a\u6570\u548cx\u4e92\u7d20\u3002
\n\\phi(x) = x\\prod(1-\\frac{1}{Pi})\uff0cPi<\/code>\u4e3ax\u7684\u8d28\u56e0\u6570\u3002\u4e00\u4e2a\u8d28\u56e0\u6570\u53ea\u8ba1\u7b97\u4e00\u6b21\u3002
\n\u6b27\u62c9\u51fd\u6570\u662f\u79ef\u6027\u51fd\u6570\u3002
\n\u7ebf\u6027\u6c42\u6b27\u62c9\u51fd\u6570\uff1a<\/p>\n
void getEular(){\n    mu[1] = 1, cnt = 0, vis[1] = 1;\n    for (ll i=2; i<=n; i++){\n        if (!vis[i])    prime[++cnt] = i, mu[i] = -1, eular[i] = i-1;\n        for (int j=1; j<=cnt && i*prime[j]<=n; j++){\n            vis[i*prime[j]] = 1;\n            if (i%prime[j]==0) {\n                eular[i*prime[j]] = eular[i]*prime[j];\n                break;\n            }else   mu[i*prime[j]] = -mu[i], eular[i*prime[j]] = eular[i]*eular[prime[j]];\n        }\n    }\n    for (int i=1; i<=n; i++)    mu[i] += mu[i-1];\n}<\/code><\/pre>\n
\u5bf9\u4e8ei%prime[j]==0<\/code>\u8fd9\u79cd\u60c5\u51b5\u7684\u8bc1\u660e\uff0c\u56e0\u4e3aprime[j]\u662fi\u6700\u5c0f\u7684\u8d28\u56e0\u6570\u3002\u6211\u4eec\u6839\u636e\u552f\u4e00\u5206\u89e3\u5b9a\u7406\u628ai\u8868\u793a\u6210n=\\prod_{i=1}^{m}p_{i}^{q_{i}}<\/code>\u7684\u5f62\u5f0f\uff0c\u7136\u540e\u5f00\u59cb\u63a8\u7b97\uff1a\\phi(n) = \\prod_{i=1}^{m}\\phi(p_{i}^{q_{i}}) = \\prod_{i=1}^{m}p_{i}^{q_{i}} - p_{i}^{q_{i}-1}(\u4e0d\u4e0ep_{i}^{q_{i}}\u4e92\u8d28\u7684\u53ea\u6709p_i\u7684\u500d\u6570\uff0c\u6709p_i^{q_i-1}\u4e2a) =  \\prod_{i=1}^{m}p_{i}^{q_{i-1}}(p_i-1)<\/code>\uff0c\u6839\u636e\u4e0a\u5f0f\uff0c\u67d0\u4e2a\u8d28\u56e0\u6570\u7684\u5e42\u591a\u4e00\u6b21\uff0cn\u7684\u6b27\u62c9\u51fd\u6570\u503c\u5c31\u4f1a\u4e58\u4ee5\u8fd9\u4e2a\u8d28\u56e0\u6570\uff0c\u8bc1\u6bd5\u3002
\n\u6309\u7167\u4e0a\u8ff0\u601d\u8def\uff0c\u53ef\u4ee5\u7528\u7ebf\u6027\u7b5b\u6c42\u5f97\u4e00\u7c7b\u79ef\u6027\u51fd\u6570\uff0c\u5373\u4e3b\u8981\u4ece3\u4e2a\u6b65\u9aa4\u6765\u6c42\u3002\u5047\u8bbe\u8981\u6c42\u79ef\u6027\u51fd\u6570f(x)<\/code>\uff1a
\n\u7b2c\u2460\u6b65\uff1a\u5f53x\u4e3a\u8d28\u6570\u65f6\uff0c\u76f4\u63a5\u6c42\uff1b
\n\u7b2c\u2461\u6b65\uff1a\u5f53x\u4e3a\u975e\u8d28\u6570\u65f6\uff0c\u53ef\u4ee5\u628ax\u8868\u793a\u6210x=i*prime[j]<\/code>\u7684\u5f62\u5f0f\uff0c\uff0cprime[j]<\/code>\u662fx\u7684\u6700\u5c0f\u8d28\u56e0\u5b50\u3002\u82e5i%prime[j]==0<\/code>\uff0c\u8bf4\u660ei\u7684\u6700\u5c0f\u8d28\u56e0\u5b50\u4e5f\u662fprime[j]<\/code>\uff0c\u6b64\u65f6\u6839\u636e\u552f\u4e00\u5206\u89e3\u5b9a\u7406\uff1af(x)=f(\\prod p_i^{q_i}) = \\prod f(p_i^{q_i})<\/code>\uff0c\u76f8\u5f53\u4e8e\u5728i\u7684\u6700\u5c0f\u8d28\u56e0\u5b50\u7684\u5e42\u4e0a\u591a\u4e861\uff0c\u8ba8\u8bba\u5206\u6790\u6b64\u65f6f(x)<\/code>\u7684\u503c\u4e3a\u4f55\u3002\u8fd9\u4e00\u6b65\u901a\u5e38\u6765\u8bf4\u662f\u6bd4\u8f83\u9ebb\u70e6\u7684\u4e00\u6b65\u3002
\n\u7b2c\u2462\u6b65\uff1a\u5728\u2461\u4e2d\uff0c\u5f53i%prime[j]!=0<\/code>\u65f6\uff0cgcd(i, prime[j])=1<\/code>\uff0ci\u6ca1\u6709prime[j]<\/code>\u8fd9\u4e2a\u8d28\u56e0\u5b50\uff0c\u6839\u636e\u79ef\u6570\u51fd\u6570\u6027\u8d28\u6709f(x) = f(i)*f(prime[j])<\/code>\u3002<\/p>\n
\u6570\u8bba\u5206\u5757<\/h4>\n
\u5bf9\u4e8e\u4e00\u4e2a\u6574\u6570n\uff0cn\u9664\u4ee5[1,n]\u5185\u7684\u6240\u6709\u6574\u6570d\u5411\u4e0b\u53d6\u6574\uff0c\u603b\u5171\u67092\\sqrt(n)<\/code>\u79cd\u53d6\u503c\u3002<\/p>\n
\u5bf9\u4e8ed<=\\sqrt(n)<\/code>\uff0c\\lfloor \\frac{n}{d} \\rfloor<\/code>\u5171\u6709\\sqrt(n)<\/code>\u79cd\u53d6\u503c\uff1b
\n\u5bf9\u4e8ed>=\\sqrt(n)<\/code>\uff0c\\lfloor \\frac{n}{d} \\rfloor<\/code>\u5171\u6709\\sqrt(n)<\/code>\u79cd\u53d6\u503c\u3002<\/p>\n
\u5bf9\u4e8e\u4e00\u4e2a\u6570d\uff0cn\u9664\u4ee5\u533a\u95f4[d, \\lfloor \\frac{n}{\\lfloor \\frac{n}{d}  \\rfloor} \\rfloor]<\/code>\u5185\u7684\u6570\u5411\u4e0b\u53d6\u6574\u5f97\u5230\u7684\u503c\u7ed3\u679c\u90fd\u662f\u4e00\u6837\u7684\u3002<\/p>\n
\u72c4\u5229\u514b\u96f7\u5377\u79ef<\/h4>\n
\u8bbef\uff0cg\u662f\u4e24\u4e2a\u51fd\u6570\uff0c\u5b83\u4eec\u7684\u72c4\u5229\u514b\u96f7\u5377\u79ef\u662f(f*g)(n) = \\sum_{d|n} f(d)g(\\frac{n}{d})<\/code>\uff0c\u5c31\u662fn\u7684\u7ea6\u6570\u5bf9\u5206\u522b\u53d6f\uff0cg\u51fd\u6570\u76f8\u4e58\u5728\u76f8\u52a0\u5c31\u884c\u4e86\u3002<\/p>\n
\u72c4\u5229\u514b\u96f7\u5377\u79ef\u6ee1\u8db3\u4ea4\u6362\u5f8b\uff0c\u7ed3\u5408\u5f8b\u3002<\/p>\n
\u5e76\u4e14\u5728\u7528\u5230\u7684\u6700\u591a\u7684\u51e0\u4e2a\u51fd\u6570\u4e2d\u6709\u5982\u4e0b\u6027\u8d28\uff1a
\n\\phi * I = id<\/code>
\n\\mu * I = \\varepsilon<\/code>
\n\\mu * id = \\phi<\/code><\/p>\n
\u4e00\u5b9a\u8981\u719f\u8bb0\u8fd9\u4e9b\u4e2a\u51fd\u6570\uff1a\\phi, \\mu, ... \\varepsilon, I, id<\/code><\/p>\n
\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14\u516c\u5f0f<\/h4>\n
\u8bbef(n), F(n)<\/code>\u4e3a\u4e24\u4e2a\u6570\u8bba\u516c\u5f0f\u3002
\n\u5982\u679c\u6709F(n) = \\sum_{n|d}f(d)<\/code>\uff0cd\u4e3an\u7684\u500d\u6570\uff0c\u5219\u6709f(n) = \\sum_{n|d} \\mu(\\frac{d}{n}) F(d)<\/code>
\n\u5982\u679c\u6709F(n) = \\sum_{d|n}f(d)<\/code>\uff0cd\u4e3an\u7684\u7ea6\u6570\uff0c\u5219\u6709f(n) = \\sum_{d|n} \\mu(\\frac{n}{d}) F(d)<\/code><\/p>\n
\u8fd9\u4e24\u4e2a\u516c\u5f0f\u7684\u7406\u8bba\u652f\u6301\u662f\uff1a\\mu * I = \\varepsilon<\/code><\/p>\n
\u5728\u4ee5\u4e0a\u7406\u8bba\u57fa\u7840\u4e0a\uff0c\u83ab\u6bd4\u4e4c\u65af\u53cd\u6f14\u7684\u9898\u76ee\u4e00\u822c\u90fd\u662f\u7ecf\u8fc7\u4e00\u7cfb\u5217\u63a8\u5f0f\u5b50\u4e4b\u540e\uff0c\u53ef\u4ee5\u5148\u7ebf\u6027\u6216\u975e\u7ebf\u6027\u9884\u5904\u7406\uff0c\u7136\u540e\u5728\u53ef\u4ee5\u5206\u5757\u5728sqrt(n)<\/code>\u65f6\u95f4\u590d\u6742\u5ea6\u5185\u89e3\u51b3\u5355\u4e2a\u6d4b\u8bd5\u6837\u4f8b\u3002\u4e00\u822c\u96be\u70b9\u662f\u63a8\u5f0f\u5b50\u548c\u9884\u5904\u7406\u3002<\/p>\n
\u603b\u7ed3\u7684\u6027\u8d28<\/h4>\n
\\phi(ij) = \\frac{\\phi(i) \\phi(j) gcd(i, j)}{\\phi(gcd(i,j))}<\/code>\uff0c\u7279\u6b8a\u7684\uff0c\u5f53j\u4e3ai\u7684\u56e0\u5b50\u65f6\uff0c\\phi(ij) = j\\phi(i)<\/code>\uff0c\u8fd9\u4e2a\u6027\u8d28\u53ef\u4ee5\u7528\u552f\u4e00\u5206\u89e3\u5b9a\u7406\u548c\u79ef\u6027\u51fd\u6570\u7684\u6027\u8d28\u5f88\u5bb9\u6613\u5f97\u5230\u3002
\nd(ij) = \\sum_{x|i} \\sum_{y|j}[gcd(x, y)=1]<\/code>\u7ea6\u6570\u4e2a\u6570\u51fd\u6570\u5206\u89e3\u3002
\n\u4e00\u822c\u7684\u7b80\u5355\u9898\u76ee\u63a8\u5230\u6700\u540e\u5c31\u662f\u5206\u5757+\u79ef\u6027\u51fd\u6570\u6c42\u548c\u95ee\u9898\uff0c\u79ef\u6027\u51fd\u6570\u8981\u4e48\u7ebf\u7b5b\uff0c\u8981\u4e48\u76f4\u63a5nlnn<\/code>\uff0c\u8981\u4e48\u5c31\u662f\u8303\u56f4\u6bd4\u8f83\u5927\uff0c\u7528\u675c\u6559\u7b5b\u6c42\u524d\u7f00\uff1b\u4e5f\u53ef\u80fd\u662f\u5206\u5757\u642d\u914d\u4e00\u4e9b\u4e0d\u80fd\u76f4\u63a5\u6c42\u7684\u51fd\u6570\uff0c\u9700\u8981\u63a8\u5bfc\u6765\u6c42\u5176\u524d\u7f00\u548c\u3002\u5f53\u7136\u8fd8\u6709\u66f4\u96be\u7684\u4e0d\u4f1a\u505a...<\/p>\n
\u4e0b\u9762\u63a8\u51e0\u4e2a\u6bd4\u8f83\u7ecf\u5178\u7684\u5f0f\u5b50\uff0c\u5957\u8def\u5c31\u5728\u91cc\u9762\u3002\u3002
\n\"\"<\/p>\n
\u63a8\u6f141<\/h6>\n
\"\u63a8\u6f141\"<\/p>\n
\u63a8\u6f142<\/h6>\n
\"\u63a8\u6f142\"<\/p>\n
\u63a8\u6f143<\/h6>\n
\"\u63a8\u6f143\"<\/p>\n
\u63a8\u6f144<\/h6>\n
\"\u63a8\u6f144\"<\/p>\n
\u63a8\u6f145<\/h6>\n
\"\u63a8\u6f145\"<\/p>\n
\u63a8\u6f146<\/h6>\n
\"\u63a8\u6f146\"<\/p>\n
\u9898\u76ee<\/h4>\n
\u4f59\u6570\u6c42\u548c<\/a>
\n\u6a21\u79ef\u548c<\/a>
\nGCD SUM<\/a>
\nGCD<\/a>
\nGCD - Extreme (I)<\/a>
\nGCD - Extreme (II)<\/a>
\n\u4e8e\u795e\u4e4b\u6012\u52a0\u5f3a\u7248<\/a>
\n[\u56fd\u5bb6\u96c6\u8bad\u961f]Crash\u7684\u6570\u5b57\u8868\u683c \/ JZPTAB<\/a>
\n[SDOI2015]\u7ea6\u6570\u4e2a\u6570\u548c<\/a>
\n[SDOI2017]\u6570\u5b57\u8868\u683c<\/a>
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\n[]()<\/p>\n","protected":false},"excerpt":{"rendered":"
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