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(unused_imports)]
use std::cmp;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use proconio::derive_readable;
use proconio::fastout;
#[allow(unused_imports)]
use num::Integer;
#[allow(unused_imports)]
use itertools::Itertools;

const MOD: i64 = 1_000_000_007;

#[derive(Copy, Clone, Hash, Debug)]
struct ModInt {
  x: i64
}
impl ModInt {
  fn new(x: i64) -> Self {
    let x = x % MOD;
    ModInt {
      x: if x < 0 {
        x + MOD
      } else {
        x
      }
    }
  }
  fn pow(self, e: i64) -> Self {
    if e == 0 {
      Self::new(1)
    } else if e % 2 == 1 {
      self.pow(e - 1) * self
    } else {
      let y = self.pow(e / 2);
      y * y
    }
  }
  #[allow(dead_code)]
  fn inv(self) -> Self {
    self.pow(MOD - 2)
  }
  #[allow(dead_code)]
  fn fact(n: usize) -> Vec<Self> {
    let mut ret = vec![0.into(); n+1];
    ret[0] = 1.into();
    for i in 1 .. n + 1 {
      ret[i] = ret[i-1] * i as i64;
    }
    ret
  }
  #[allow(dead_code)]
  fn inv_fact(fact: &Vec<Self>) -> Vec<Self> {
    let n = fact.len() - 1;
    let mut ret = vec![0.into(); n+1];
    ret[n] = fact[n].inv();
    for i in (0 .. n).rev() {
      ret[i] = ret[i+1] * (i + 1) as i64;
    }
    ret
  }
}
impl<T: Into<ModInt>> std::ops::Add<T> for ModInt {
  type Output = Self;
  fn add(self, other: T) -> Self {
    let other = other.into();
    let sum = self.x + other.x;
    ModInt {
      x: if sum >= MOD { sum - MOD } else { sum }
    }
  }
}
impl<T: Into<ModInt>> std::ops::Sub<T> for ModInt {
  type Output = Self;
  fn sub(self, other: T) -> Self {
    let other = other.into();
    let sum = self.x - other.x;
    ModInt {
      x: if sum < 0 { sum + MOD } else { sum }
    }
  }
}
impl<T: Into<ModInt>> std::ops::Mul<T> for ModInt {
  type Output = Self;
  fn mul(self, other: T) -> Self {
    let other = other.into();
    ModInt {
      x: self.x * other.x % MOD
    }
  }
}
impl<T: Into<ModInt>> std::ops::Div<T> for ModInt {
  type Output = Self;
  fn div(self, other: T) -> Self {
    let other = other.into();
    self * other.inv()
  }
}
impl From<i64> for ModInt {
  fn from(x: i64) -> Self {
    ModInt::new(x)
  }
}
impl std::fmt::Display for ModInt {
  fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
    self.x.fmt(f)
  }
}
impl<T: Into<ModInt>> std::ops::AddAssign<T> for ModInt {
  fn add_assign(&mut self, other: T) {
    *self = *self + other;
  }
}
impl<T: Into<ModInt>> std::ops::SubAssign<T> for ModInt {
  fn sub_assign(&mut self, other: T) {
    *self = *self - other;
  }
}
impl<T: Into<ModInt>> std::ops::MulAssign<T> for ModInt {
  fn mul_assign(&mut self, other: T) {
    *self = *self * other;
  }
}
impl<T: Into<ModInt>> std::ops::DivAssign<T> for ModInt {
  fn div_assign(&mut self, other: T) {
    *self = *self / other;
  }
}

#[fastout]
fn main() {
  input! {
    n: i64,
    k: i64,
  }
  let mut ans = ModInt::new(0);
  let mut pat = vec![ModInt::new(0); k as usize + 1];
  for g in (1 ..= k).rev() {
    let candidate = k / g;
    let mut p = ModInt::new(candidate).pow(n);
    for m in 2 .. {
      if m * g > k {
        break;
      }
      p -= pat[(m * g) as usize];
    }
    pat[g as usize] = p;
    ans += p * g;
  }

  println!("{}", ans);
}