UVa 11417 GCD (欧拉φ函数)

 

UVa 11417 GCD (欧拉φ函数)

分类： acm之路--数学 数论 UVa 2013-10-20 10:05  379人阅读  评论(0)  收藏  举报

acm UVa c++ 数学

目录(?)[+]

11417 - GCD

Time limit: 2.000 seconds

http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=24&page=show_problem&problem=2412

Given the value of N, you will have to find the value of G. The definition of G is given below:

 

Here GCD(i,j) means the greatest common divisor of integer i and integer j.

 

For those who have trouble understanding summation notation, the meaning of G is given in the following code:

G=0; for(i=1;i<N;i++) for(j=i+1;j<=N;j++) {     G+=GCD(i,j); } /*Here GCD() is a function that finds the greatest common divisor of the two input numbers*/

 

Input

The input file contains at most 100 lines of inputs. Each line contains an integer N (1<N<501). The meaning of N is given in the problem statement. Input is terminated by a line containing a single zero.  This zero should not be processed.

 

Output

For each line of input produce one line of output. This line contains the value of G for corresponding N.

 

Sample Input                 Output for Sample Input

10 100 500  0

67 13015 442011

如何求?

思路：可以直接算，复杂度O(N^2 logN)，但是我们可以找到一种复杂度更小的算法O(N loglogN)

以10为例，与之互素的有φ(10)=4个(1,3,7,9)，与之gcd=2的有φ(10/2)=4个(2,4,6,8)，与之gcd=5的有φ(10/5)=1个(5)

这样，10提供的G值就是5*φ(2)+2*φ(5)+φ(10)=5+8+4=17

根据上述计算过程可以得到如下公式：

（方括号指艾弗森约定，当方括号内语句为真时其值为1，假时为0，参见《具体数学》P21）

完整代码：

 /*0.013s*/

 #include<cstdio>
 const int maxn = 501;

 int phi[maxn], G[maxn];

 void init()
 {
     int i, j;
     for (i = 2; i < maxn; ++i)
         phi[i] = i;
     for (i = 2; i < maxn; ++i)
     {
         if (phi[i] == i)
             for (j = i; j < maxn; j += i)
                 phi[j] = phi[j] / i * (i - 1);///计算欧拉φ函数
         for (j = 1; j * i < maxn; ++j)
             G[j * i] += j * phi[i];
     }
     for (i = 3; i < maxn; ++i)
         G[i] += G[i - 1];
 }

 int main()
 {
     init();
     int n;
     while (scanf("%d", &n), n)
         printf("%d\n", G[n]);
     return 0;
 }

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