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 Rust (1.42.0) (judge verdict: Accepted)

#![allow(
    non_snake_case,
    unused_variables,
    unused_assignments,
    unused_mut,
    unused_imports,
    dead_code
)]
//use permutohedron::LexicalPermutation as _;
use proconio::input;
//use proconio::marker::*;
use std::collections::*;
//use std::cmp::*;
//use std::num;
//use petgraph::unionfind::UnionFind;
const MOD: i64 = 1_000_000_007;
fn main() {
    input! {
        N:i64,mut K:i64
    }
    let mut ans = 0;
    let mut gcd_num = vec![0i64; K as usize + 1];
    for a in (1..=K).rev() {
        gcd_num[a as usize] = modPow(K / a, N);
        for i in 2..=K / a {
            gcd_num[a as usize] = (gcd_num[a as usize] - gcd_num[(a * i) as usize]) % MOD;
        }
        ans = (ans + gcd_num[a as usize] * a % MOD) % MOD;
    }

    ans = (ans + MOD) % MOD;
    println!("{}", ans);
}

fn modPow(x: i64, a: i64) -> i64 {
    if a == 1 {
        x
    } else if a & 1 == 1 {
        x * modPow(x, a - 1) % MOD
    } else {
        modPow(x, a / 2).pow(2) % MOD
    }
}