Sigma Function (平方数与平方数*2的约数和是奇数)

Posted 2021-02-13 fighting-sh

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

 

 

 1 #include<bits/stdc++.h>
 2 using namespace std;
 3 #define lson l,mid,rt<<1
 4 #define rson mid+1,r,rt<<1|1
 5 #define sqr(x) ((x)*(x))
 6 #define pb push_back
 7 #define eb emplace_back
 8 #define maxn 100005
 9 #define eps 1e-8
10 #define pi acos(-1.0)
11 #define rep(k,i,j) for(int k=i;k<j;k++)
12 typedef long long ll;
13 typedef pair<int,int> pii;
14 typedef pair<long long,int>pli;
15 typedef pair<int,char> pic;
16 typedef pair<pair<int,string>,pii> ppp;
17 typedef unsigned long long ull;
18 const long long mod=998244353;
19 /*#ifndef ONLINE_JUDGE
20         freopen("1.txt","r",stdin);
21 #endif */
22 
23 int pow_mul(ll a,ll b){
24     int ans=1;
25     while(b){
26         if(b&1) ans=ans*a%1000;
27         b>>=1;
28         a=a*a%1000;
29     }
30     return ans;
31 }
32 
33 int main(){
34     #ifndef ONLINE_JUDGE
35      //   freopen("1.txt","r",stdin);
36     #endif
37   //  std::ios::sync_with_stdio(false);
38     int t;
39     cin>>t;
40     for(int _=1;_<=t;_++){
41         ll n;
42         cin>>n;
43         int ans=0;
44         for(ll i=1;i*i<=n;i++){
45             ans++;
46             if(i*i*2<=n) ans++;
47         }    
48         cout<<"Case "<<_<<": "<<n-ans<<endl;
49     }
50 }

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