Light OJ-----1336数论-----整数拆分推论

最新推荐文章于 2019-03-11 17:33:39 发布

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最新推荐文章于 2019-03-11 17:33:39 发布

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分类专栏：数学题/思维题

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本文介绍了一种高效算法，用于计算在1到n范围内整数的因子和为偶数的数量。通过利用数学特性，特别是关于平方数及其两倍数的因子和的奇偶性规律，该算法能够快速得出结果。

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Sigma Function

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

Output for Sample Input

100

1000

Case 1: 1

Case 2: 5

Case 3: 83

Case 4: 947

求从1到n因子和为偶数的个数

不知道为何，，，，，，

一堆公式证明翻过来倒过去就推出来平方数和平方数*2的约数和是奇数-------------------

而且-----平方数*2的个数就等于sqrt（n/2）

.........数学真神奇

#include<cstdio>
#include<cmath>
int main(){
	int t, ca = 0;
    scanf("%d", &t);
    while(t--){
		long long n, a, b, ans;
        scanf("%lld", &n);
		a = sqrt(n);
		b = sqrt(n/2);
		ans = a + b;
        printf("Case %d: %lld\n",++ca, n-ans);
    }
    return 0;
}