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
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 3000010
using 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;
}