1336 - Sigma Functio

Posted 2020-10-11 缄默。

1336 - 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.

 

分析：

对于任意一个x， 都有x = p1^a1*p2^a2*...*pn^an;所以x的所有因子和f(x) = ( 1 + p1 + p1^2+ ...p1^a1)(1 + p2 + p2 ^ 2 + ...p2^a2)...(1 + pn + pn^2 +...pn^an).
偶数和偶数乘积为偶，偶和奇的乘积为偶，只有奇和奇的乘积为奇，所以我们求出和为奇的然后减去就是偶的了。
1x只有素因子2时 加上1一定为奇。
2偶数个奇数相加为偶，只有素因子2为偶数，加上1 为奇数，所以ai需为偶数，所以完全平方数x^2的每一个p^a（a一定会是偶数，因为是两个x相乘，所以就是两个a相加，不管是奇数加奇数，还是偶数加偶数都会是偶数）
3x^2因子和是偶数了，那么2*x^2的因子和也一定是偶数。因为就算再多一个2也没关系，最后还是会加上一个1还是奇数。

所以最后只用减去2^x,x^2和2*x^2，x^2和2*x^2又包含2^x,所用只用减去x^2和2*x^2.
代码：

#include<cstdio>
#include<cstring>
#include<cmath>
#include<iostream>
#include<algorithm>
#include<queue>
#include<stack>

using namespace std;
typedef long long ll;
const int maxn = 1e7+5;
const int mod = 1000;

int quickmi(int a, int b)
{
if(b == 0)
return 1;

int tmp = quickmi(a, b>>1);

tmp = tmp * tmp % mod;

if(b & 1)
tmp = tmp * (a % mod) % mod;

return tmp % mod;

}
int main(void)
{
int T, cas;
ll n;

scanf("%d", &T);

cas = 0;

while(T--)
{
cas++;

scanf("%lld", &n);

ll sum;

sum = n;
sum -= (int)sqrt(n);
sum -= (int)sqrt(n/2);

printf("Case %d: %lld\n", cas, sum);

}

return 0;
}