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 <bits/stdc++.h>
#include <variant>
#include <boost/range/algorithm.hpp>
#include <boost/range/numeric.hpp>
#include <boost/range/irange.hpp>
#include <boost/range/adaptor/indexed.hpp>
#define FOR(i,k,n)  for(int i = (k);i < (n);++i)
#define REP(i,n)    FOR(i,0,n)
#define ALL(x) begin(x),end(x)

using namespace std;
using namespace boost::adaptors;
using namespace std::string_literals;
using ll = int64_t;
using vecint = vector<int>;
using vecll = vector<ll>;
using boost::irange;
constexpr ll MOD = 1000000007;

ll pmod(ll base, ll index) {
  if (index == 0) return 1;
  if (index%2 == 1) return pmod(base, index-1) * base % MOD;
  ll h = pmod(base, index/2);
  return h * h % MOD;
}

int main() {
  ios::sync_with_stdio(false);
  cin.tie(nullptr);
  ll n,k;
  cin>>n>>k;
  vecll v(k+1);
  FOR(i,1,k+1) {
    v[i] = pmod(k/i, n);
  }
  for (int i = k; i > 0; --i) {
    for (int j = 2*i; j <= k; j += i) {
      v[i] += MOD - v[j];
      v[i] %= MOD;
    }
  }
  ll ans = 0;
  FOR(i,1,k+1) {
    ans += i * v[i] % MOD;
    ans %= MOD;
  }
  cout<<ans<<endl;
  return 0;
}