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: Accepted)

def gcd(x,y):
    while(y): 
        x, y = y, x % y 
    return x
    
MOD = 10 ** 9 + 7

out = 0
N, K = map(int, input().split())

val = [i for i in range(K+1)]
for i in range(2,K+1):
    if val[i] == i:
        for j in range(i, K + 1, i):
            val[j] //= i
            val[j] *= (i - 1)

for i in range(1, K + 1):
    count = K//i
    out += val[i] * pow(count, N, MOD)
    out %= MOD
print(out)