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: Time Limit Exceeded)

(v=>{
    const n = ~~(v.split(' ')[0])
    const k = ~~(v.split(' ')[1])
    const gcd = (x,y) =>{
        const f = (a,b) => b ? f(b, a % b): a 
        return f(x,y)
    }
    // 前計算
    const pre = []
    for(let i=0;i<k; ++i){
        const preadd = {}
        for(let j=0;j<k; ++j){
            const g = gcd(i+1, j+1)
            if(preadd[g] === undefined)preadd[g] =0
            preadd[g] += 1
        }
        const preadd2 = []
        Object.keys(preadd).forEach(key=>{
            preadd2.push([+key,preadd[key]])
        },preadd)
        pre.push(preadd2) 
    }
    
    const dp = [new Array(k+1).fill(1)]

    for(let i=1; i< n; ++i){
        dp.push(new Array(k+1).fill(0))
        for(let j=0; j<k; ++j){
            for(let preelm of pre[j]){
                dp[i][preelm[0]-1] +=  dp[i-1][j] * preelm[1]
                dp[i][preelm[0]-1] %= 1e+9+7
            }
        }
    }

    const last_dp = dp.pop()
    let res = 0
    for(let i=0; i<last_dp.length; ++i){
        res += (i+1) * last_dp[i]
        res %= 1e+9+7
    }
    console.log(res)

})(require("fs").readFileSync("/dev/stdin", "utf8"))