LightOJ - 1336 - Sigma Function（质数分解）

链接:

https://vjudge.net/problem/LightOJ-1336

题意：

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.

思路：

考虑质数的约数和可以看作\((1+p_1^1+p_1^2+...+p_1^{k1})*(1+p_2^1+p_2^2+...+p_2^{k2})...\)

考虑和为奇数情况。

存在素因子2时，（1+2^n)为奇数，则和为奇数。

其他素因子为奇数，奇数乘奇数为奇数，1+偶数为奇数，偶数乘奇数为偶数，则只有素因子的指数为偶数时满足和为奇数。

由此得到其他数都为平方数。或者时平方数*2

代码：

#include<iostream>
#include<cstdio>
#include<cstring>
#include<string>
#include<algorithm>
#include<math.h>
#include<vector>
#include<map>

using namespace std;
typedef long long LL;
const int INF = 1e9;

const int MAXN = 1e6+10;
const int MOD = 1e9+7;

LL n;

int main()
{
    int t, cnt = 0;
    scanf("%d", &t);
    while(t--)
    {
        printf("Case %d:", ++cnt);
        scanf("%lld", &n);
        LL tmp = n;
        tmp -= (LL)sqrt(n);
        tmp -= (LL)sqrt(n/2);
        printf(" %lld\n", tmp);
    }

    return 0;
}