LightOJ
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题目:
Mathematically some problems look hard. But with the help of the computer, some problems can be easily solvable.
In this problem, you will be given two integers a and b. You have to find the summation of the scores of the numbers from a to b (inclusive). The score of a number is defined as the following function.
score (x) = n2, where n is the number of relatively prime numbers with x, which are smaller than x
For example,
For 6, the relatively prime numbers with 6 are 1 and 5. So, score (6) = 22 = 4.
For 8, the relatively prime numbers with 8 are 1, 3, 5 and 7. So, score (8) = 42 = 16.
Now you have to solve this task.
Input
Input starts with an integer T (≤ 105), denoting the number of test cases.
Each case will contain two integers a and b (2 ≤ a ≤ b ≤ 5 * 106).
Output
For each case, print the case number and the summation of all the scores from a to b.
Sample Input
3
6 6
8 8
2 20
Sample Output
Case 1: 4
Case 2: 16
Case 3: 1237
Hint
Euler’s totient function applied to a positive integer n is defined to be the number of positive integers less than or equal to n that are relatively prime to n. is read “phi of n.”
Given the general prime factorization of , one can compute using the formula
题意:
输入T,表示测试组数。输入a,b。表示区间。要将区间a到b的所有数的欧拉数的平方加起来输出。
解题思路:
因为数据较大,直接套公式挨个求解,肯定超时,这个时候就需要根据欧拉函数性质打表了。 φ(p) = p – 1 (p为素数) 若m,n互质,φ(nm) = φ(m) * φ(n)。至于代码稍后给出。如果只是简单打表,这个题还是会超时,需要进一步优化。这个时候就需要前缀和了。前缀和指把当前项到开始第一项都加起来。用处在代码中可以体现。如果只是这样,还是会wa。因为数据太大。即使是long long也会溢出。那么这个时候就需要无符号长整型了。AC.
代码:
//欧拉函数 打表 前缀和 无符号long long # include <cstdio># include <cstring># define ll unsigned long long //定义ll代表无符号整型using namespace std;ll phi[5000002];void oula(ll n) //欧拉函数打表{ memset(phi,0,sizeof(phi)); phi[1] = 1; for(int i = 2;i <= n;i++) { if(phi[i] == 0) { for(int j = i;j <= n;j+=i) { if(phi[j] == 0) phi[j] = j; phi[j] = phi[j]/i * (i-1); } } }} int main(){ int T; int l,r; int p = 1; oula(5000001); for(int i = 2;i <= 5000000;i++) //依照规则求前缀和 { phi[i] = phi[i-1] + phi[i] * phi[i]; } scanf("%d",&T); while(T--) { scanf("%d %d",&l,&r); //因为前缀和求出,相减就行,这样时间就大大优化 printf("Case %d: %llu\n",p++,phi[r] - phi[l-1]); } return 0;}
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