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  • [SDOi2012]Longge的问题

    嘟嘟嘟


    .

    [egin{align*} ans &= sum_{i = 1} ^ {n} (i, n) \ &= sum _ {d | n} ^ {n} sum _ {(i, n) = d} ^ {n} d \ &= sum _ {d | n} ^ {n} sum _ {(i, frac{n}{d}) = 1} ^ {frac{n}{d}} d \ &= sum _ {d | n} ^ {n} d * varphi(frac{n}{d}) end{align*}]

    然后(O(sqrt{n}))枚举(n)的因数,(O(sqrt{n}))求欧拉函数即可。时间复杂度为(O()因数个数$ * sqrt{n})$。

    #include<cstdio>
    #include<iostream>
    #include<cmath>
    #include<algorithm>
    #include<cstring>
    #include<cstdlib>
    #include<cctype>
    #include<vector>
    #include<stack>
    #include<queue>
    using namespace std;
    #define enter puts("") 
    #define space putchar(' ')
    #define Mem(a, x) memset(a, x, sizeof(a))
    #define rg register
    typedef long long ll;
    typedef double db;
    const int INF = 0x3f3f3f3f;
    const db eps = 1e-8;
    //const int maxn = ;
    inline ll read()
    {
      ll ans = 0;
      char ch = getchar(), last = ' ';
      while(!isdigit(ch)) last = ch, ch = getchar();
      while(isdigit(ch)) ans = (ans << 1) + (ans << 3) + ch - '0', ch = getchar();
      if(last == '-') ans = -ans;
      return ans;
    }
    inline void write(ll x)
    {
      if(x < 0) x = -x, putchar('-');
      if(x >= 10) write(x / 10);
      putchar(x % 10 + '0');
    }
    
    ll phi(ll n)
    {
      ll ret = n;
      for(ll i = 2; i * i <= n; ++i)
        if(n % i == 0)
          {
    	ret = ret / i * (i - 1);
    	while(n % i == 0) n /= i;
          }
      if(n > 1) ret = ret / n * (n - 1);
      return ret;
    }
    ll solve(ll n)
    {
      ll ret = 0, i = 1;
      for(; i * i < n; ++i)
        if(n % i == 0) ret += n / i * phi(i) + i * phi(n / i);
      if(i * i == n) ret += i * phi(i);
      return ret;
    }
    
    int main()
    {
      ll n = read();
      write(solve(n)), enter;
      return 0;
    }
    
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  • 原文地址:https://www.cnblogs.com/mrclr/p/10108040.html
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