Sigma Function 数学因子求和

最新推荐文章于 2025-08-27 11:49:14 发布

转载最新推荐文章于 2025-08-27 11:49:14 发布 · 102 阅读

0 ·

CC 4.0 BY-SA版权

原文链接：http://www.cnblogs.com/joeylee97/p/6829730.html

本文探讨了数论中的Sigma函数，介绍了其定义及计算方法，特别关注于寻找那些Sigma值为偶数的整数数量。文章提供了一种有效算法，用于解决在大范围内寻找满足条件的数的问题。

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<iostream>
#include<cstdio>
#include<cstring>
#include<sstream>
#include<algorithm>
#include<queue>
#include<vector>
#include<cmath>
#include<map>
#include<stack>
#include<set>
#include<fstream>
#include<memory>
#include<list>
#include<string>
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
#define MAXN  1000010
#define LLL 1000000000
#define INF 1000000009
/*
给一个公式，将一个数表示为所有素因子等比数列 求前e个 的乘积
求小于n的数中为奇数的个数
数据很大不能打表。。
当p为偶数 ,p的次方为偶数，加上p的0次方 最后得出奇数不成立
当P为奇数，P的次方为奇数，要有偶数个奇数相加才能得出偶数，要求e为奇数
当所有的 p和e中总是存在任何一个不为奇数的时候，结果为奇数
e为偶数 平方数  后面乘某些数 后面的数字是2的几次方的时候仍然成立 （2是唯一的素偶数）
所有个数为 平方数||2*平方数
*/
int main()
{
    LL T,n;
    scanf("%lld", &T);
    for(LL cas=1;cas<=T;cas++)
    {
        scanf("%lld", &n);
        LL t1 = sqrt(n*1.0);
        LL t2 = sqrt(n*0.5);
        printf("Case %lld: %lld\n",cas, n - t1 - t2);
    }
    return 0;
}

转载于:https://www.cnblogs.com/joeylee97/p/6829730.html

Sigma Function 数学 因子求和

Sigma Function 数学因子求和