poj 2480 Longge's problem（欧拉函数的性质）

Longge's problem

Time Limit: 1000MS		Memory Limit: 65536K
Total Submissions: 6433		Accepted: 2064

Description
Longge is good at mathematics and he likes to think about hard mathematical problems which will be solved by some graceful algorithms. Now a problem comes: Given an integer N(1 < N < 2^31),you are to calculate ∑gcd(i, N) 1<=i <=N.

"Oh, I know, I know!" Longge shouts! But do you know? Please solve it.

Input
Input contain several test case.

A number N per line.

Output
For each N, output ,∑gcd(i, N) 1<=i <=N, a line
Sample Input

2
6

Sample Output

3
15

Source
POJ Contest,Author:Mathematica@ZSU

题意：求 ∑gcd(i, N) 1<=i <=N
题解：涉及欧拉函数，积极函数，莫比乌斯反演等...本弱菜无法说明，搞了很久，但是代码非常短的说.....这大概就是典型的数论题的特征吧....

#include<stdio.h>
int main()
{
long long n,temp,res,i,cou;

while(scanf("%I64d",&n)>0)
{
for(res=1,i=2;i*i<=n;i++)
{
if(n%i!=0) continue;
cou=0;  temp=1;
while(n%i==0){ n/=i; temp*=i; cou++; }
res*=temp/i*(i-1)*cou+temp;
}
if(n>1) res*=2*n-1;
printf("%I64d\n",res);
}

return 0;
}