The 36th ACM/ICPC Asia Regional Dalian Site —— Online Contest Find the maximum

   
   
   
   

    
    
    
    http://acm.hdu.edu.cn/showproblem.php?pid=4002
    
    
    
     
   
   
   
   
   
   
   
   

    
    
    
     
    
    
    
    
 
     Problem Description 
    
    
    
    
    
 
     Euler's Totient function, φ (n) [sometimes called the phi function], is used to determine the number of numbers less than n which are relatively prime to n . For example, as 1, 2, 4, 5, 7, and 8, are all less than nine and relatively prime to nine, φ(9)=6. HG is the master of X Y. One day HG wants to teachers XY something about Euler's Totient function by a mathematic game. That is HG gives a positive integer N and XY tells his master the value of 2<=n<=N for which φ(n) is a maximum. Soon HG finds that this seems a little easy for XY who is a primer of Lupus, because XY gives the right answer very fast by a small program. So HG makes some changes. For this time XY will tells him the value of 2<=n<=N for which n/φ(n) is a maximum. This time XY meets some difficult because he has no enough knowledge to solve this problem. Now he needs your help.  
    
    
    
    
    
 
       
    
    
    
    
     
    
    
    
    
 
     Input 
    
    
    
    
    
 
     There are T test cases (1<=T<=50000). For each test case, standard input contains a line with 2 ≤ n ≤ 10^100.  
    
    
    
    
    
 
       
    
    
    
    
     
    
    
    
    
 
     Output 
    
    
    
    
    
 
     For each test case there should be single line of output answering the question posed above.  
    
    
    
    
    
 
       
    
    
    
    
     
    
    
    
    
 
     Sample Input 
    
    
    
    
    
 
     
      
      
      
      

       
       
       
       2 10 100
      
      
      
      
 
    
    
    
    
    
 
       
    
    
    
    
    

    
    
    
    
 
     Sample Output 
    
    
    
    
    
 
     
      
      
      
      

       
       
       
       6 30 
      
      
      
      
 
    
    
    
    
    题意是：给你一个N,2 <= n <=N，求n/φ(n)最大时n的值。
分析：φ(n)=n(1-1/p1)(1-1/p2)(1-1/p3)(1-1/p4)…..(1-1/pk)欧拉函数:对正整数n，
    
    
    
    欧拉
    
    
    
    函数是少于或等于n的数中与n互质的数的数目。
n/φ(n)=(p1/(p1-1))(p2/(p2-1))(p3/(p3-1))(p4/(p4-1))......(pk/(pk-1));
因此素数因子越多
    
    
    
    n/φ(n)越大，特别注意：要保证素因子积小于N。
    
    
    
    
   
   
   
   
   
   
   
   

    
    
    
    java：
   
   
   
   
   
   
   
   

    
    
    
    
    
    
    
    
import java.math.*;
import java.util.*;
public class Main{
	  public static void main(String[]args){
		   Scanner in=new Scanner(System.in);
		   int num=1000;
		   boolean[] b=new boolean[num+1];
		   long[] bb=new long[num];
		   for(int i=0;i<num;i++) b[i]=true;
		    for(int i=2;i*i<num+1;i++)
		    	  for(int j=i*i;j<num+1;j+=i)
		    		    b[j]=false;
		    int k=0;
		    for(int i=1;i<num+1;i++) 
		    	if(b[i]) bb[k++]=i;//素数打表
		       int n=in.nextInt();
		       BigInteger N;
		           while(n--!=0)
		        {   BigInteger m=BigInteger.ONE;
		            N=in.nextBigInteger();
		            for(int i=0;m.compareTo(N)==-1;i++)
		          {  if(m.multiply(BigInteger.valueOf(bb[i])).compareTo(N)>0) break;
		                 m=m.multiply(BigInteger.valueOf(bb[i]));
		            }
		            System.out.println(m);
		        	}
		 }
	  
}