Codeforces 839D Winter is here【数学：容斥原理】

D. Winter is here

time limit per test：3 seconds

memory limit per test：256 megabytes

input：standard input

output：standard output

Winter is here at the North and the White Walkers are close. John Snow has an army consisting of n soldiers. While the rest of the world is fighting for the Iron Throne, he is going to get ready for the attack of the White Walkers.

He has created a method to know how strong his army is. Let the i-th soldier’s strength be a_i. For some k he calls i₁, i₂, ..., i_k a clan if i₁ < i₂ < i₃ < ... < i_k and gcd(a_i1, a_i2, ..., a_ik) > 1 . He calls the strength of that clan k·gcd(a_i1, a_i2, ..., a_ik). Then he defines the strength of his army by the sum of strengths of all possible clans.

Your task is to find the strength of his army. As the number may be very large, you have to print it modulo 1000000007 (10⁹ + 7).

Greatest common divisor (gcd) of a sequence of integers is the maximum possible integer so that each element of the sequence is divisible by it.

Input

The first line contains integer n (1 ≤ n ≤ 200000) — the size of the army.

The second line contains n integers a₁, a₂, ..., a_n (1 ≤ a_i ≤ 1000000) — denoting the strengths of his soldiers.

Output

Print one integer — the strength of John Snow's army modulo 1000000007 (10⁹ + 7).

Examples

Input

3
3 3 1

Output

12

Input

4
2 3 4 6

Output

39

Note

In the first sample the clans are {1}, {2}, {1, 2} so the answer will be 1·3 + 1·3 + 2·3 = 12

题目链接：http://codeforces.com/contest/839/problem/D

下面给出AC代码：

#include<cstdio>
const int N=1000010,P=1000000007;
int n,i,j,x,ans,po[N],a[N],f[N];
int main(){
  scanf("%d",&n);
  for(po[0]=i=1;i<=n;i++)po[i]=2*po[i-1]%P;
  while(n--)scanf("%d",&x),a[x]++;
  for(i=N-1;i>1;i--){
    for(j=i,x=0;j<N;j+=i)x+=a[j];
    if(x){
      f[i]=1LL*x*po[x-1]%P;
      for(j=i+i;j<N;j+=i)f[i]=(f[i]-f[j]+P)%P;
      ans=(1LL*f[i]*i+ans)%P;
    }
  }
  printf("%d",ans);
}