Problem:
Score : 500 points

Problem Statement

Consider sequences \{A_1,...,A_N\} of length N consisting of integers between 1 and K (inclusive).

There are K^N such sequences. Find the sum of \gcd(A_1, ..., A_N) over all of them.

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

Here \gcd(A_1, ..., A_N) denotes the greatest common divisor of A_1, ..., A_N.

Constraints

2 \leq N \leq 10^5

1 \leq K \leq 10^5

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N K

Output

Print the sum of \gcd(A_1, ..., A_N) over all K^N sequences, modulo (10^9+7).

Sample Input 1

3 2

Sample Output 1

9

\gcd(1,1,1)+\gcd(1,1,2)+\gcd(1,2,1)+\gcd(1,2,2)
+\gcd(2,1,1)+\gcd(2,1,2)+\gcd(2,2,1)+\gcd(2,2,2)
=1+1+1+1+1+1+1+2=9

Thus, the answer is 9.

Sample Input 2

3 200

Sample Output 2

10813692

Sample Input 3

100000 100000

Sample Output 3

742202979

Be sure to print the sum modulo (10^9+7).

Solution in JavaScript (Node.js 12.16.1) (judge verdict: Wrong Answer)


"use strict";

const main = arg => {
    var count = 0
  const input = arg.trim().split("\n");
  const N = parseInt(input[0].split(" ")[0]);
  const K = parseInt(input[0].split(" ")[1]);

  var a = K;
  for (var s = 1;s<=N;s++){
      a *= K;
  }
  console.log(a);


     for(var i = 1;i <= a;i++){
    for(var j = 1;j <= a;j++){
     for(var k = 1;k <= a;k++){
        count+=gcd(i,j,k);   
  }
  }
  }
  console.log(count/10000000007);
}
 
function gcd() {
    var f = (a, b) => b ? f(b, a % b) : a
    var ans = arguments[0]
    for (var i = 1; i < arguments.length; i++) {
        ans = f(ans, arguments[i]); 
    }
    return ans 
}
main(require('fs').readFileSync('/dev/stdin', 'utf8'));