A11500. Coprime Arrays

Let'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 $10^{9}+7$ .

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.

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).

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. $b_{4}=55$ .