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 itertools::Itertools;
#[allow(unused_imports)]
use num::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;

#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct ModInt(u32);

impl ModInt {
  pub const MOD: u32 = 1_000_000_007;

  pub fn inv(self) -> Self {
    if self.0 == 0 {
      panic!();
    }
    self.pow(Self::MOD - 2)
  }

  pub fn pow(self, e: u32) -> Self {
    if e == 0 {
      return Self::new(1);
    }
    let mut res = self.pow(e >> 1);
    res *= res;
    if e & 1 == 1 {
      res *= self;
    }
    res
  }

  fn new(n: i64) -> Self {
    let mut n = n % (Self::MOD as i64);
    if n.is_negative() {
      n += Self::MOD as i64;
    }
    Self(n as u32)
  }
}

impl From<i32> for ModInt {
  fn from(n: i32) -> Self {
    ModInt::from(n as i64)
  }
}

impl From<i64> for ModInt {
  fn from(n: i64) -> Self {
    Self::new(n)
  }
}

impl From<isize> for ModInt {
  fn from(n: isize) -> Self {
    ModInt::from(n as i64)
  }
}

impl From<u32> for ModInt {
  fn from(n: u32) -> Self {
    ModInt::from(n as u64)
  }
}

impl From<u64> for ModInt {
  fn from(n: u64) -> Self {
    Self::new(n as i64)
  }
}

impl From<usize> for ModInt {
  fn from(n: usize) -> Self {
    ModInt::from(n as u64)
  }
}

impl Into<i32> for ModInt {
  fn into(self) -> i32 {
    self.0 as i32
  }
}

impl Into<i64> for ModInt {
  fn into(self) -> i64 {
    self.0 as i64
  }
}

impl Into<isize> for ModInt {
  fn into(self) -> isize {
    self.0 as isize
  }
}

impl Into<u32> for ModInt {
  fn into(self) -> u32 {
    self.0
  }
}

impl Into<u64> for ModInt {
  fn into(self) -> u64 {
    self.0 as u64
  }
}

impl Into<usize> for ModInt {
  fn into(self) -> usize {
    self.0 as usize
  }
}

impl std::fmt::Display for ModInt {
  fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
    write!(f, "{}", self.0)
  }
}

impl std::ops::Add for ModInt {
  type Output = Self;

  fn add(self, rhs: Self) -> Self {
    Self::new((self.0 + rhs.0) as i64)
  }
}

impl std::ops::AddAssign for ModInt {
  fn add_assign(&mut self, rhs: Self) {
    *self = *self + rhs;
  }
}

impl std::ops::Div for ModInt {
  type Output = Self;

  fn div(self, rhs: Self) -> Self {
    self * rhs.inv()
  }
}

impl std::ops::DivAssign for ModInt {
  fn div_assign(&mut self, rhs: Self) {
    *self = *self / rhs;
  }
}

impl std::ops::Mul for ModInt {
  type Output = Self;

  fn mul(self, rhs: Self) -> Self {
    Self::new((self.0 as i64) * (rhs.0 as i64))
  }
}

impl std::ops::MulAssign for ModInt {
  fn mul_assign(&mut self, rhs: Self) {
    *self = *self * rhs;
  }
}

impl std::ops::Sub for ModInt {
  type Output = Self;

  fn sub(self, rhs: Self) -> Self {
    Self::new((self.0 as i64) - (rhs.0 as i64))
  }
}

impl std::ops::SubAssign for ModInt {
  fn sub_assign(&mut self, rhs: Self) {
    *self = *self - rhs;
  }
}

fn main() {
  input! {
    n: usize, k: usize
  };
  let mut counts = vec![ModInt::from(0); k + 1];
  for a in 1..=k {
    counts[a] = ModInt::from(k / a).pow(n as u32);
  }
  let mut res = ModInt::from(0);
  for a in (1..=k).rev() {
    res += ModInt::from(a) * counts[a];
    let mut m = 2;
    let mut c = ModInt::from(0);
    while a * m <= k {
      c += counts[a * m];
      m += 1;
    }
    counts[a] -= c;
    res -= ModInt::from(a) * c;
  }
  println!("{}", res);
}