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 Kotlin (1.3.71) (judge verdict: Accepted)

import java.io.BufferedReader
import java.io.InputStream
import java.io.InputStreamReader
import java.util.*
import kotlin.math.abs
import kotlin.math.max
import kotlin.math.min

val MOD = 1_000_000_007

class Comb(n: Int, val mod: Int) {

  val F = LongArray(n + 1)
  val I = LongArray(n + 1)
  init {
    F[0] = 1
    for (i in 1..n) {
      F[i] = F[i - 1] * i % mod
    }
    I[n] = powMod(F[n], (mod - 2).toLong(), mod)
    for (i in n - 1 downTo 0) {
      I[i] = I[i + 1] * (i + 1) % mod
    }
  }

  private fun powMod(a: Long, n: Long, mod: Int): Long {
    if (n == 0L) return 1
    val res = powMod(a * a % mod, n / 2, mod)
    return if (n % 2 == 1L) res * a % mod else res
  }

  fun comb(n: Int, k: Int): Long {
    return F[n] * I[k] % mod * I[n - k] % mod
  }

  fun inv(x: Int): Long {
    return I[x] * F[x - 1] % mod
  }

  /**
   * nのグループからk回重複ありで選ぶ組み合わせ数
   * n - 1のしきりとkの○で考える
   */
  fun H(n: Int, k: Int) = comb(n + k - 1, k)
}
fun powMod(a: Int, n: Long, mod: Int): Long {
  if (n == 0L) return 1
  val res = powMod((a.toLong() * a % mod).toInt(), n / 2, mod)
  return if (n % 2 == 1L) res * a % mod else res
}


class Solver(stream: InputStream, private val out: java.io.PrintWriter) {
  fun solve() {
    val N = nl()
    val K = ni()
    var ans = 0L
    val dp = LongArray(K + 10)
    for (g in 1 .. K) {
      val m = K / g
      val v= (MOD + g - dp[g]) % MOD
      val cmb = powMod(m, N, MOD)
      ans = (ans + cmb * v % MOD) % MOD
      var x = g
      while(x <= K) {
        dp[x] = (dp[x] + v) % MOD
        x += g
      }
    }

    out.println(ans)
  }













































  private val isDebug = try {
    // なんか本番でエラーでる
    System.getenv("MY_DEBUG") != null
  } catch (t: Throwable) {
    false
  }

  private var tokenizer: StringTokenizer? = null
  private val reader = BufferedReader(InputStreamReader(stream), 32768)
  private fun next(): String {
    while (tokenizer == null || !tokenizer!!.hasMoreTokens()) {
      tokenizer = StringTokenizer(reader.readLine())
    }
    return tokenizer!!.nextToken()
  }

  private fun ni() = next().toInt()
  private fun nl() = next().toLong()
  private fun ns() = next()
  private fun na(n: Int, offset: Int = 0): IntArray {
    return map(n) { ni() + offset }
  }
  private fun nal(n: Int, offset: Int = 0): LongArray {
    val res = LongArray(n)
    for (i in 0 until n) {
      res[i] = nl() + offset
    }
    return res
  }

  private fun na2(n: Int, offset: Int = 0): Array<IntArray> {
    val a  = Array(2){IntArray(n)}
    for (i in 0 until n) {
      for (e in a) {
        e[i] = ni() + offset
      }
    }
    return a
  }

  private inline fun map(n: Int, f: (Int) -> Int): IntArray {
    val res = IntArray(n)
    for (i in 0 until n) {
      res[i] = f(i)
    }
    return res
  }

  private inline fun debug(msg: () -> String) {
    if (isDebug) System.err.println(msg())
  }

  private fun debug(a: LongArray) {
    debug { a.joinToString(" ") }
  }

  private fun debug(a: IntArray) {
    debug { a.joinToString(" ") }
  }

  private fun debug(a: BooleanArray) {
    debug { a.map { if (it) 1 else 0 }.joinToString("") }
  }

  private fun debugDim(A: Array<IntArray>) {
    if (isDebug) {
      for (a in A) {
        debug(a)
      }
    }
  }

  /**
   * 勝手にimport消されるのを防ぎたい
   */
  private fun hoge() {
    min(1, 2)
    max(1, 2)
    abs(-10)
  }
}

fun main() {
  val out = java.io.PrintWriter(System.out)
  Solver(System.`in`, out).solve()
  out.flush()
}