Sum of gcd of Tuples (Hard)

AtCoder

IDabc162_e

Time2000ms

Memory256MB

Difficulty—

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: $N$ $K$ [samples]

Samples

Input #1

3 2

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 $9$.

Input #2

3 200

Output #2

10813692

Input #3

100000 100000

Output #3

742202979

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

API Response (JSON)

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