A11500 | Coprime Arrays
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内存限制256MB
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题目描述
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 $10^{9}+7$ .
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 $10^{9}+7$ .
输入格式
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.
输出格式
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).
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).
输入输出样例
输入 #1
3 4
输出 #1
82
输入 #2
2000000 8
输出 #2
339310063
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. $b_{4}=55$ .
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. $b_{4}=55$ .
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评测结果：Accepted