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 Python (3.8.2) (judge verdict: Accepted)

n,k = map(int,input().split())

mod = 10**9+7

all_num = pow(k,n,mod)
#all_num = k**n


ans = 0

dp = [0]*(k+1)

for i in range(k):
    tmp = k-i
    #print(tmp)
    #print(tmp,tmp,tmp)
    kazu = 1
    kakeru = 2
    heru = 0
    
    if tmp == 1:
        ans += all_num
        ans %= mod
        break
    
    #print(tmp,k)
    for i in range(k//tmp-1):
        #print(i)
        heru += dp[tmp*kakeru]
        kazu += 1
        kakeru += 1
    """
    while tmp * kakeru <= k:
        heru += dp[tmp*kakeru]
        kazu += 1
        kakeru += 1
        #print(tmp*kakeru,k)
    """ 
    
    kazu = pow(kazu,n,mod)-heru
    #kazu = kazu**n-heru
    ans += tmp*(kazu)
    all_num -= (kazu)
    #print(tmp,kazu,n,heru)
    dp[tmp] = kazu
    ans %= mod
    #print(ans,tmp,kazu)
    
print(ans)