Light OJ-----1336数论-----整数拆分推论

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

4

3

10

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;}