Sigma Function (平方数与平方数*2的约数和是奇数)

Sigma Function

https://vjudge.net/contest/288520#problem/D

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

4

3

10

100

1000

Sample Output

Case 1: 1

Case 2: 5

Case 3: 83

Case 4: 947

 

 

#include<bits/stdc++.h>
using namespace std;
#define lson l,mid,rt<<1
#define rson mid+1,r,rt<<1|1
#define sqr(x) ((x)*(x))
#define pb push_back
#define eb emplace_back
#define maxn 100005
#define eps 1e-8
#define pi acos(-1.0)
#define rep(k,i,j) for(int k=i;k<j;k++)
typedef long long ll;
typedef pair<int,int> pii;
typedef pair<long long,int>pli;
typedef pair<int,char> pic;
typedef pair<pair<int,string>,pii> ppp;
typedef unsigned long long ull;
const long long mod=998244353;
/*#ifndef ONLINE_JUDGE
        freopen("1.txt","r",stdin);
#endif */

int pow_mul(ll a,ll b){
    int ans=1;
    while(b){
        if(b&1) ans=ans*a%1000;
        b>>=1;
        a=a*a%1000;
    }
    return ans;
}

int main(){
    #ifndef ONLINE_JUDGE
     //   freopen("1.txt","r",stdin);
    #endif
  //  std::ios::sync_with_stdio(false);
    int t;
    cin>>t;
    for(int _=1;_<=t;_++){
        ll n;
        cin>>n;
        int ans=0;
        for(ll i=1;i*i<=n;i++){
            ans++;
            if(i*i*2<=n) ans++;
        }    
        cout<<"Case "<<_<<": "<<n-ans<<endl;
    }
}