Virtual contest is a way to take part in past contest, as close as possible to participation on time. It is supported only ACM-ICPC mode for virtual contests.
If you've seen these problems, a virtual contest is not for you - solve these problems in the archive.
If you just want to solve some problem from a contest, a virtual contest is not for you - solve this problem in the archive.
Never use someone else's code, read the tutorials or communicate with other person during a virtual contest.

No tag edit access

G. Coprime Arrays

time limit per test

3.5 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputLet's call an array *a* of size *n* coprime iff *gcd*(*a*_{1}, *a*_{2}, ..., *a*_{n}) = 1, where *gcd* is the greatest common divisor of the arguments.

You are given two numbers *n* and *k*. For each *i* (1 ≤ *i* ≤ *k*) you have to determine the number of coprime arrays *a* of size *n* such that for every *j* (1 ≤ *j* ≤ *n*) 1 ≤ *a*_{j} ≤ *i*. Since the answers can be very large, you have to calculate them modulo 10^{9} + 7.

Input

The first line contains two integers *n* and *k* (1 ≤ *n*, *k* ≤ 2·10^{6}) — the size of the desired arrays and the maximum upper bound on elements, respectively.

Output

Since printing 2·10^{6} numbers may take a lot of time, you have to output the answer in such a way:

Let *b*_{i} be the number of coprime arrays with elements in range [1, *i*], taken modulo 10^{9} + 7. You have to print , taken modulo 10^{9} + 7. Here denotes bitwise xor operation (^ in C++ or Java, xor in Pascal).

Examples

Input

3 4

Output

82

Input

2000000 8

Output

339310063

Note

Explanation of the example:

Since the number of coprime arrays is large, we will list the arrays that are non-coprime, but contain only elements in range [1, *i*]:

For *i* = 1, the only array is coprime. *b*_{1} = 1.

For *i* = 2, array [2, 2, 2] is not coprime. *b*_{2} = 7.

For *i* = 3, arrays [2, 2, 2] and [3, 3, 3] are not coprime. *b*_{3} = 25.

For *i* = 4, arrays [2, 2, 2], [3, 3, 3], [2, 2, 4], [2, 4, 2], [2, 4, 4], [4, 2, 2], [4, 2, 4], [4, 4, 2] and [4, 4, 4] are not coprime. *b*_{4} = 55.

Codeforces (c) Copyright 2010-2018 Mike Mirzayanov

The only programming contests Web 2.0 platform

Server time: Apr/24/2018 21:07:11 (d1).

Desktop version, switch to mobile version.

User lists

Name |
---|