uva 11526 H(n) (数论)

时间:2023-01-28 11:58:26

转载自 http://blog.csdn.net/synapse7/article/details/12873437

这道题我自己做的时候没有想到好的优化方法,提交的时候借鉴这篇文章的内容,转载如下:

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11526 - H(n)

Time limit: 5.000 seconds

http://uva.onlinejudge.org/index.php?option=com_onlinejudge&Itemid=8&category=24&page=show_problem&problem=2521

What is the value this simple C++ function will return?

long long H(int n){

      long long res = 0;

     for( int i = 1; i <= n; i=i+1 ){

            res = (res + n/i);

      }

     return res;

}

 

Input
The first line of input is an integer T ( T <= 1000 ) that indicates the number of test cases. Each of the next T line will contain a single signed 32 bit integer n.

 

Output
For each test case, output will be a single line containing H(n).

Sample Input                      Output for Sample Input

2

5

10

 

10

27

怎么计算sum{ [n/i] }?(1<=i<=n)(n<=2147483647)

n太大,硬算肯定不行,我们先观察一个例子,看能否得出一些结论。

当n=20时,和式展开为

20+10+6+5+4+3+2+2+2+2+1+1+1+1+1+1+1+1+1+1

注意到后面相同的数太多,不妨化简下:

20+10+6+5+1*(20-10)+2*(10-6)+3*(6-5)+4*(5-4)

=(20+10+6+5)+(20+10+6+5)-4*4

=2(20+10+6+5)-4*4

也许,我们可以

uva 11526 H(n) (数论)

这样,复杂度就从O(n)降为O(√n)了。