#P915G. Coprime Arrays
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ID: 6422
Type: RemoteJudge
3000ms
256MiB
Tried: 0
Accepted: 0
Difficulty: (None)
Uploaded By:
Hydro
Tags> Coprime Arrays
No submission language available for this problem.
Description
Let's call an array a of size n coprime iff gcd(a1, a2, ..., an) = 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 ≤ aj ≤ i. Since the answers can be very large, you have to calculate them modulo 109 + 7.
The first line contains two integers n and k (1 ≤ n, k ≤ 2·106) — the size of the desired arrays and the maximum upper bound on elements, respectively.
Since printing 2·106 numbers may take a lot of time, you have to output the answer in such a way:
Let bi be the number of coprime arrays with elements in range [1, i], taken modulo 109 + 7. You have to print 
, taken modulo 109 + 7. Here 
denotes bitwise xor operation (^ in C++ or Java, xor in Pascal).
Input
The first line contains two integers n and k (1 ≤ n, k ≤ 2·106) — the size of the desired arrays and the maximum upper bound on elements, respectively.
Output
Since printing 2·106 numbers may take a lot of time, you have to output the answer in such a way:
Let bi be the number of coprime arrays with elements in range [1, i], taken modulo 109 + 7. You have to print , taken modulo 109 + 7. Here denotes bitwise xor operation (^ in C++ or Java, xor in Pascal).
Samples
3 4
82
2000000 8
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. b1 = 1.
For i = 2, array [2, 2, 2] is not coprime. b2 = 7.
For i = 3, arrays [2, 2, 2] and [3, 3, 3] are not coprime. b3 = 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. b4 = 55.