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 PyPy3 (7.3.0) (judge verdict: Wrong Answer)

import sys
sys.setrecursionlimit(10000000)

def gcd(x, y):
    while (y):
        x, y = y, x % y
    return x
MOD = int(10e9 + 7)



n, k = map(int, input().split())
mem = {}
def dp(at, cur):
    if at == n:
        return cur
    if (at,cur) in mem:
        return mem[(at,cur)]
    sm = 0
    for i in range(1, k+1):
        sm += dp(at+1,gcd(cur, i))
    mem[(at,cur)] = sm % MOD
    return sm % MOD

sm = 0
for i in range(1, k+1):
    sm += dp(1, i)
print(sm % MOD)