Discription
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
首先发现2这个质因子没有用，所以我们就枚举每个数的2的次数是多少，然后/2^这个次数之后钦定它是奇数。
我们发现满足条件的数必须质因子里有至少一个的次数是奇数，所以直接补集转化减一下就行了。
#include<bits/stdc++.h>
#define ll long long
using namespace std;
int T;
ll ans,N;
inline ll g(ll x){ return (x+1)>>1;}
inline ll f(ll x){ return g(x)-g((ll)floor(sqrt(x+0.5)));}
int main(){
scanf("%d",&T);
for(int i=1;i<=T;i++){
ans=0,scanf("%lld",&N);
while(N) ans+=f(N),N>>=1;
printf("Case %d: %lld\n",i,ans);
}
return 0;
}