hdu 2824 The Euler function(欧拉函数)

The Euler function

Time Limit: 2000/1000 MS (Java/Others)    Memory Limit: 32768/32768 K (Java/Others)
Total Submission(s): 6640    Accepted Submission(s): 2775

Problem Description

The Euler function phi is an important kind of function in number theory, (n) represents the amount of the numbers which are smaller than n and coprime to n, and this function has a lot of beautiful characteristics. Here comes a very easy question: suppose you are given a, b, try to calculate (a)+ (a+1)+....+ (b)

 

Input

There are several test cases. Each line has two integers a, b (2<a<b<3000000).

 

Output

Output the result of (a)+ (a+1)+....+ (b)

 

Sample Input

3 100

 

Sample Output

3042

 

Source

2009 Multi-University Training Contest 1 - Host by TJU

 

题意：（a）是与a互质且小于a的值的个数。

定义：对于正整数n，φ(n)是小于或等于n的正整数中，与n互质的数的数目。
　　　　例如：φ(8)=4，因为1，3，5，7均和8互质。
性质：1.若p是质数，φ(p)= p-1.
　　　2.若n是质数p的k次幂，φ(n)=(p-1)*p^(k-1)。因为除了p的倍数都与n互质
　　　3.欧拉函数是积性函数，若m,n互质，φ(mn)= φ(m)φ(n).

在程序中利用欧拉函数如下性质，可以快速求出欧拉函数的值（a为N的质因素）

　　若( N%a ==0&&(N/a)%a ==0)则有：E(N)= E(N/a)*a;
　　若( N%a ==0&&(N/a)%a !=0)则有：E(N)= E(N/a)*(a-1);

#include<cstdio>#include <algorithm>  #include <cstring> #define LL long long#define N 3000010using namespace std;int a,b;int phi[N],prime[N],fprime[N];  //fprime是标记函数，fprime[i]为0时说明i为质数。prime为质数数组，用以保存质数void Get(){int ans,k=0;memset(fprime,0,sizeof(fprime));for(int i=2;i<N;i++){if(fprime[i]==0){prime[k++]=i;phi[i]=i-1;}for(int j=0;j<k&&i*prime[j]<N;j++){fprime[i*prime[j]]=1;if(i%prime[j]==0)   phi[i*prime[j]]=phi[i]*prime[j];else    phi[i*prime[j]]=phi[i]*(prime[j]-1);}}}int main(){Get();while(~scanf("%d%d",&a,&b)){LL sum=0;for(int i=a;i<=b;i++)   sum+=phi[i];printf("%lld\n",sum);} return 0;}