hdu1299Diophantus of Alexandria
1/x+1/y = 1/n 设y = n + k;==>1/x + 1/(n+k)=1/n;==>x = n^2/k + n;因为x为整数，k就是n^2的约数。————题解

Problem Description
Diophantus of Alexandria was an egypt mathematician living in Alexandria. He was one of the first mathematicians to study equations where variables were restricted to integral values. In honor of him, these equations are commonly called diophantine equations. One of the most famous diophantine equation is x^n + y^n = z^n. Fermat suggested that for n > 2, there are no solutions with positive integral values for x, y and z. A proof of this theorem (called Fermat's last theorem) was found only recently by Andrew Wiles.
``````#include <iostream>
#include<cstdio>
#include<cstring>
using namespace std;
const int N=1e7+9;
long long prime[N/10]={0},num_prime=0;
bool isnotprime[N];
int fac[100000];
void init()
{
memset(isnotprime,false,sizeof(isnotprime));
isnotprime[0]=true,isnotprime[1]=true;
for(long long i=2;i<N;i++)
{
if(!isnotprime[i])
prime[num_prime++]=i;
for(long long j=0;j<num_prime&&i*prime[j]<N;j++)
{
isnotprime[i*prime[j]]=true;
if(!(i%prime[j]))break;
}
}
}
long long get(int x)
{
int tot=0;
memset(fac,0,sizeof(fac));
for(int i=0;i<num_prime&&prime[i]*prime[i]<=x;i++)
{
if(x%prime[i]==0)
{
while(x%prime[i]==0)
fac[tot]++,x/=prime[i];
tot++;
}
}
if(x>1) fac[tot++]=1;
long long ans=1;
for(int i=0;i<tot;i++)
ans=ans*(2*fac[i]+1);
return ans;
}
int main()
{
// freopen("cin.txt","r",stdin);
init();
int t,cas=1;
scanf("%d",&t);
while(t--){
long long n;
scanf("%lld",&n);
printf("Scenario #%d:\n%I64d\n\n",cas++,(get(n)+1)/2);
}
return 0;
}
``````

Consider the following diophantine equation:

<center> 1 / x + 1 / y = 1 / n where x, y, n ∈ N+ (1) </center>

Diophantus is interested in the following question: for a given n, how many distinct solutions (i. e., solutions satisfying x ≤ y) does equation (1) have? For example, for n = 4, there are exactly three distinct solutions:

<center> 1 / 5 + 1 / 20 = 1 / 4
1 / 6 + 1 / 12 = 1 / 4
1 / 8 + 1 / 8 = 1 / 4 </center>

Clearly, enumerating these solutions can become tedious for bigger values of n. Can you help Diophantus compute the number of distinct solutions for big values of n quickly?

Input
The first line contains the number of scenarios. Each scenario consists of one line containing a single number n (1 ≤ n ≤ 10^9).

Output
The output for every scenario begins with a line containing "Scenario #i:", where i is the number of the scenario starting at 1. Next, print a single line with the number of distinct solutions of equation (1) for the given value of n. Terminate each scenario with a blank line.

Sample Input
2 4 1260

Sample Output
Scenario #1: 3 Scenario #2: 113

2022-12-31 23:19

2022-12-23 14:45

2022-12-30 11:36

2022-12-09 14:30

2022-12-16 11:59

2022-12-08 11:32

01-06 21:26

01-04 15:28

2022-12-08 17:43

2022-12-08 15:09