1245 - Harmonic Number (II)（规律题）
1245 - Harmonic Number (II)
| Time Limit: 3 second(s) | Memory Limit: 32 MB |
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 |
Output for Sample Input |
|
11 1 2 3 4 5 6 7 8 9 10 2147483647 |
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，自己有思路，却写不出来，看了答案，倒是挺简单的；
先看两个例子
1.
n = 10 sqrt(10) = 3 10/sqrt(10) = 3
i 1 2 3 4 5 6 7 8 9 10
n/i 10 5 3 2 2 1 1 1 1 1
m = n/i
sum += m;
m = 1的个数10/1-10/2 = 5；
m = 2的个数10/2-10/3 = 2；
m = 3的个数10/3-10/4 = 1；
2.
n = 20 sqrt(20) = 4 20/sqrt(20) = 5
i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
n/i 20 10 6 5 4 3 2 2 2 2 1 1 1 1 1 1 1 1 1 1
m = n/i
sum += m;
m = 1的个数20/1-20/2 = 10；
m = 2的个数20/2-20/3 = 4；
m = 3的个数20/3-20/4 = 1；
m = 4的个数20/4-20/5 = 1；
...
m = i的个数20/i - 20/(i + 1)(1<= i <= sqrt(n))
这样我们可以得出：sqrt(n)之前的数我们可以直接用for循环来求
sqrt(n)之后的sum += (n/i - n/(i + 1)) * i;
当sqrt(n) = n / sqrt(n)时(如第一个例子10，sum就多加了一个3)，sum多加了一个sqrt(n),减去即可;
无语，这里输出%I64d竟然wa。。。
代码：
1 #include<iostream> 2 #include<cstdio> 3 #include<cstring> 4 #include<cmath> 5 #include<algorithm> 6 #define mem(x,y) memset(x,y,sizeof(x)) 7 using namespace std; 8 typedef long long LL; 9 const int INF=0x3f3f3f3f; 10 11 int main(){ 12 int T; 13 int n,cnt=0; 14 LL res; 15 scanf("%d",&T); 16 while(T--){ 17 res=0; 18 scanf("%d",&n); 19 int m=sqrt(n); 20 for(int i=1;i<=m;i++)res+=n/i; 21 for(int i=1;i<=m;i++) 22 res+=(n/i-n/(i+1))*i; 23 if(m==n/m)res-=m; 24 printf("Case %d: %lld\n",++cnt,res); 25 } 26 return 0; 27 }
