E - Sum of gcd of Tuples (Hard)

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 $(10^9+7)$.

3 2

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 $9$.

3 200

10813692

100000 100000

742202979

Be sure to print the sum modulo $(10^9+7)$.