F - Sigma Function

最新推荐文章于 2024-12-21 16:50:21 发布

WXR1999

最新推荐文章于 2024-12-21 16:50:21 发布

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本文介绍了数论中有趣的Sigma函数，该函数表示一个数的所有除数之和，并提供了一个计算公式。文章还给出了一段C++代码，用于计算1到n之间具有偶数Sigma值的整数数量。

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Sigma function is an interesting function in Number Theory. It is denoted by the Greek letter Sigma (σ). This function actually denotes the sum of all divisors of a number. For example σ(24) = 1+2+3+4+6+8+12+24=60. Sigma of small numbers is easy to find but for large numbers it is very difficult to find in a straight forward way. But mathematicians have discovered a formula to find sigma. If the prime power decomposition of an integer is

Then we can write,

For some n the value of σ(n) is odd and for others it is even. Given a value n, you will have to find how many integers from 1 to n have even value of σ.

Input

Input starts with an integer T (≤ 100), denoting the number of test cases.

Each case starts with a line containing an integer n (1 ≤ n ≤ 1012).

Output

For each case, print the case number and the result.

Sample Input

100

1000

Sample Output

Case 1: 1

Case 2: 5

Case 3: 83

Case 4: 947

#include <cstring>
#include <stdio.h>
#include <algorithm>
#include <math.h>
#define MAXN 1e7+5
using namespace std;
typedef long long LL;

int main()
{
    int T;
	LL n;
	scanf("%d",&T);
    while(T--)
	{
		int cas=1;
		scanf("%lld",&n);
		LL num1,num2;
		num1=(LL)sqrt((double)n);
		num2=(LL)sqrt((double)n/2.0);
        printf("Case %d: %lld\n",cas++,n-num1-num2);
	}
	return 0;
}