LightOJ 1336 Sigma Function

1336 - Sigma Function

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Time Limit: 2 second(s)Memory Limit: 32 MB

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

 

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PROBLEM SETTER: SHAHRIAR MANZOOR

SPECIAL THANKS: JANE ALAM JAN (SOLUTION, DATASET)

题意：求1到N有多少个数的因子和是偶数。

思路：很明显直接暴力，肯定会超时，那么肯定存在规律，不过没看懂题上给的算式是干什么的。

最开始计算了1到16的因子和，发现完全平方数和2的幂的数的因子和是偶数，但是不对，再往后写发现18（坑。。。。）也是奇数，即完全平方数以及他的二倍是奇数

#include<stdio.h>#include<string.h>#include<algorithm>#include<math.h>#define LL long longusing namespace std;int main(){int t,cnt=1;LL n,m,i,ans;scanf("%d",&t);while(t--){scanf("%lld",&n);ans=0;for(i=1;i*i<=n;i++){ans++;if(i*i*2<=n)ans++;}printf("Case %d: %lld\n",cnt++,n-ans);}return 0; }