Problem:
Score : 500 points

Problem Statement

Given are N positive integers A_1,...,A_N.

Consider positive integers B_1, ..., B_N that satisfy the following condition.

Condition: For any i, j such that 1 \leq i < j \leq N, A_i B_i = A_j B_j holds.

Find the minimum possible value of B_1 + ... + B_N for such B_1,...,B_N.

Since the answer can be enormous, print the sum modulo (10^9 +7).

Constraints

1 \leq N \leq 10^4

1 \leq A_i \leq 10^6

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
A_1 ... A_N

Output

Print the minimum possible value of B_1 + ... + B_N for B_1,...,B_N that satisfy the condition, modulo (10^9 +7).

Sample Input 1

3
2 3 4

Sample Output 1

13

Let B_1=6, B_2=4, and B_3=3, and the condition will be satisfied.

Sample Input 2

5
12 12 12 12 12

Sample Output 2

5

We can let all B_i be 1.

Sample Input 3

3
1000000 999999 999998

Sample Output 3

996989508

Print the sum modulo (10^9+7).

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Time Limit Exceeded)

import java.math.BigInteger;
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.Map.Entry;
import java.util.Scanner;

public class Main {
	static int PN = 1000000;
	static List<Integer> prime = new ArrayList<Integer>();
	static Map<Integer,Boolean> primeMap = new HashMap<Integer,Boolean>();
	static Map<Integer,Map<Integer,Integer>> exp = new HashMap<Integer, Map<Integer,Integer>>();
	
	public static void main(String[] args) {
		init();
		Scanner sc = new Scanner(System.in);
		int N = sc.nextInt();
		
		HashMap<Integer, Integer> map = new HashMap<Integer, Integer>();
		for(int i=1;i<=N;i++) {
			int A = sc.nextInt();
			Map<Integer,Integer> tmpMap = compute(map,A);
			exp.put(i, tmpMap);
		}
		BigInteger t = BigInteger.ONE;
		for(Entry<Integer,Integer> entry:map.entrySet()) {
			int a = entry.getKey();
			int p = entry.getValue();
			t = BigInteger.valueOf(a).pow(p).multiply(t);
		}
//		System.out.println(map);
//		System.out.println(t);
		BigInteger res = BigInteger.ZERO;
		for(int i=1;i<=N;i++) {
			Map<Integer,Integer> tmpMap = exp.get(i);
			BigInteger once = t.add(BigInteger.ZERO);
			for(Entry<Integer,Integer> entry:tmpMap.entrySet()) {
				int a = entry.getKey();
				int p = entry.getValue();
				
				once = once.divide(BigInteger.valueOf(a).pow(p));
			}
			res = res.add(once).mod(BigInteger.valueOf(1000000007));
		}
		System.out.println(res);
	}
	
	private static Map<Integer,Integer> compute(HashMap<Integer, Integer> map,int A) {
		Map<Integer,Integer> tmpMap = new HashMap<Integer, Integer>();
		for(int i=0,size=prime.size();i<size;) {
			
			if( primeMap.get(A) != null) {

				Integer count = tmpMap.get(A) ;
				
				if( count ==  null) {
					tmpMap.put(A, 1);
				}else {
					tmpMap.put(A,count+1);
				}
				
				break;
			}
			
			else if( A % prime.get(i) == 0) {
				int p =  prime.get(i);
				
				Integer count = tmpMap.get(p) ;
				
				if( count ==  null) {
					tmpMap.put(p, 1);
				}else {
					tmpMap.put(p,count+1);
				}
				
				A = A / p;
			}else {
				i++;
			}
		}
		for(Entry<Integer, Integer> entry:tmpMap.entrySet()) {
			int p = entry.getKey();
			int count = entry.getValue();
			
			Integer currCount = map.get(p);
			
			if( currCount == null || currCount <= count) {
				map.put(p,count);
			}
		}
		return tmpMap;
	}
	
	
	
	private static boolean isPrime(int n) {
		int t = (int)Math.sqrt(n);
		for(int i=2;i<=t;i++) {
			if( n% i == 0) {
				return false;
			}
		}
		return true;
	}
	
	public static void init() {
		primeMap.put(1,true);
		for(int i=2;i<=PN;i++) {
			if( isPrime(i)) {
				prime.add(i);
				primeMap.put(i, true);
			}
		}
//		System.out.println(prime.size()+";largest:"+prime.get(prime.size()-1));
	}
}