1103 Integer Factorization (30 分)
 

The KP factorization of a positive integer N is to write N as the sum of the P-th power of K positive integers. You are supposed to write a program to find the KP factorization of N for any positive integers N, K and P.

Input Specification:

Each input file contains one test case which gives in a line the three positive integers N (≤400), K (≤N) and P (1<P≤7). The numbers in a line are separated by a space.

Output Specification:

For each case, if the solution exists, output in the format:

N = n[1]^P + ... n[K]^P

where n[i] (i = 1, ..., K) is the i-th factor. All the factors must be printed in non-increasing order.

Note: the solution may not be unique. For example, the 5-2 factorization of 169 has 9 solutions, such as 12​2​​+4​2​​+2​2​​+2​2​​+1​2​​, or 11​2​​+6​2​​+2​2​​+2​2​​+2​2​​, or more. You must output the one with the maximum sum of the factors. If there is a tie, the largest factor sequence must be chosen -- sequence { a​1​​,a​2​​,⋯,aK​​ } is said to be larger than { b​1​​,b​2​​,⋯,bK​​ } if there exists 1≤LK such that ai​​=bi​​ for i<L and aL​​>bL​​.

If there is no solution, simple output Impossible.

Sample Input 1:

169 5 2

Sample Output 1:

169 = 6^2 + 6^2 + 6^2 + 6^2 + 5^2

Sample Input 2:

169 167 3

Sample Output 2:

Impossible
#include<bits/stdc++.h>
using namespace std; const int maxn=; int factor[maxn]; int n,k,p; vector<int> ans,temp; int cnt=; bool flag=false; int maxSum = -; void init(){
int i=,temp=;
while(temp<=n){
factor[i]=(int)pow(i*1.0,p);
temp=factor[i];
cnt=i;
i++;
}
} void DFS(int index,int nowk,int sumW,int sumC){
if(nowk>k||sumC>n)
return; if(nowk==k&&sumC==n){ flag=true; if(sumW>maxSum){
maxSum=sumW;
ans=temp;
}
} if(index->=){
temp.push_back(index);
DFS(index,nowk+,sumW+index,sumC+factor[index]);
temp.pop_back();
DFS(index-,nowk,sumW,sumC); } } int main(){ ios::sync_with_stdio(false);
cin.tie(); cin>>n>>k>>p; init(); DFS(cnt,,,); if(!flag){
cout<<"Impossible\n";
return ;
} cout<<n<<" ="; for(int i=;i<ans.size();i++){
if(i>)
cout<<" +"; cout<<" "<<ans[i]<<"^"<<p;
} cout<<endl; return ; }

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