Harmonic Number (II) [数学]

最新推荐文章于 2019-07-11 12:39:20 发布

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最新推荐文章于 2019-07-11 12:39:20 发布

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分类专栏：数学

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本文介绍了一种解决1234-HarmonicNumber问题的有效算法，通过优化代码实现O(√n)的时间复杂度。文章提供了完整的C语言代码示例，并给出了输入输出样例。

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I was trying to solve problem ‘1234 - Harmonic Number’, I wrote the following code

long long H( int n ) {
    long long res = 0;
    for( int i = 1; i <= n; i++ )
        res = res + n / i;
    return res;
}

Yes, my error was that I was using the integer divisions only. However, you are given n, you have to find H(n) as in my code.

Input

Input starts with an integer T (≤ 1000), denoting the number of test cases.

Each case starts with a line containing an integer n (1 ≤ n < 231).

Output

For each case, print the case number and H(n) calculated by the code.

Sample Input

11
1
2
3
4
5
6
7
8
9
10
2147483647

Sample Output

Case 1: 1
Case 2: 3
Case 3: 5
Case 4: 8
Case 5: 10
Case 6: 14
Case 7: 16
Case 8: 20
Case 9: 23
Case 10: 27
Case 11: 46475828386

题解

发现一下规律就可以写出 $o(\sqrt n )$ 的代码了

#include<stdio.h>
typedef long long LL;

int main()
{
    int T;LL n;
    scanf("%d",&T);
    for(int t=1;t<=T;t++){
        scanf("%lld",&n);
        LL l=2,r=n,ans=0;
        while(l<=n/l){
            ans+=(r-n/l)*(l-1);
            r=n/l;
            ans+=n/l;
            l++;
        }
        for(LL i=l;i<=r;i++) ans+=n/l;
        printf("Case %d: %lld\n",t,ans+n);
    }
    return 0;
}