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 C++ (GCC 9.2.1) (judge verdict: Accepted)

#include <iostream>
#include <algorithm>
#define flush fflush(stdout)
using namespace std;
typedef long long ll;


int main(void) {
    int n, k, i, j, d, de;
    const int mod = 1e9 + 7;
    ll pa[100003] = {}, r, ans;

    cin >> n >> k;
    
    for (i = k; i > 0; i--) {
        d = k / i;
        de = n;
        pa[i] = 1;
        r = d;
        while (de > 0) {
            if ((de & 1) == 1) pa[i] = pa[i] * r % mod;
            r = r * r % mod;
            de >>= 1;
        }
        j = i;
        while (j <= k) {
            if (j + i <= k) j += i;
            else break;
            pa[i] = (pa[i] + mod - pa[j]) % mod;
        }

    }

    ans = 0;
    for (i = 1; i <= k; i++) {
        ans = (ans + pa[i] * i % mod) % mod;
    }

    cout << ans << endl;

    return 0;
}