Complex Market Analysis

题意翻译

给定长为 $n$ 的正整数序列 $a$ 和一个正整数 $e$，求满足以下规定的有序数对 $(i,k)$ 的数量： - $i,k\ge 1$ - $i+e\times k\le n$ - $\prod\limits_{j=0}^ka_{i+e\times j}$ 是质数 多组数据，$1\le t\le 10^4,1\le e\le n\le 2\times 10^5,1\le a_i\le 10^6,1\le \sum n\le 2\times 10^5$。

题目描述

While performing complex market analysis William encountered the following problem: For a given array $ a $ of size $ n $ and a natural number $ e $ , calculate the number of pairs of natural numbers $ (i, k) $ which satisfy the following conditions: - $ 1 \le i, k $ - $ i + e \cdot k \le n $ . - Product $ a_i \cdot a_{i + e} \cdot a_{i + 2 \cdot e} \cdot \ldots \cdot a_{i + k \cdot e} $ is a prime number. A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.

输入输出格式

输入格式

Each test contains multiple test cases. The first line contains the number of test cases $ t $ ( $ 1 \le t \le 10\,000 $ ). Description of the test cases follows. The first line of each test case contains two integers $ n $ and $ e $ $ (1 \le e \le n \le 2 \cdot 10^5) $ , the number of items in the array and number $ e $ , respectively. The second line contains $ n $ integers $ a_1, a_2, \dots, a_n $ $ (1 \le a_i \le 10^6) $ , the contents of the array. It is guaranteed that the sum of $ n $ over all test cases does not exceed $ 2 \cdot 10^5 $ .

输出格式

For each test case output the answer in the following format: Output one line containing the number of pairs of numbers $ (i, k) $ which satisfy the conditions.

输入输出样例

输入样例 #1

6
7 3
10 2 1 3 1 19 3
3 2
1 13 1
9 3
2 4 2 1 1 1 1 4 2
3 1
1 1 1
4 1
1 2 1 1
2 2
1 2

输出样例 #1

2
0
4
0
5
0

说明

In the first example test case two pairs satisfy the conditions: 1. $ i = 2, k = 1 $ , for which the product is: $ a_{2} \cdot a_{5} = 2 $ which is a prime number. 2. $ i = 3, k = 1 $ , for which the product is: $ a_{3} \cdot a_{6} = 19 $ which is a prime number. In the second example test case there are no pairs that satisfy the conditions. In the third example test case four pairs satisfy the conditions: 1. $ i = 1, k = 1 $ , for which the product is: $ a_{1} \cdot a_{4} = 2 $ which is a prime number. 2. $ i = 1, k = 2 $ , for which the product is: $ a_{1} \cdot a_{4} \cdot a_{7} = 2 $ which is a prime number. 3. $ i = 3, k = 1 $ , for which the product is: $ a_{3} \cdot a_{6} = 2 $ which is a prime number. 4. $ i = 6, k = 1 $ , for which the product is: $ a_{6} \cdot a_{9} = 2 $ which is a prime number. In the fourth example test case there are no pairs that satisfy the conditions. In the fifth example test case five pairs satisfy the conditions: 1. $ i = 1, k = 1 $ , for which the product is: $ a_{1} \cdot a_{2} = 2 $ which is a prime number. 2. $ i = 1, k = 2 $ , for which the product is: $ a_{1} \cdot a_{2} \cdot a_{3} = 2 $ which is a prime number. 3. $ i = 1, k = 3 $ , for which the product is: $ a_{1} \cdot a_{2} \cdot a_{3} \cdot a_{4} = 2 $ which is a prime number. 4. $ i = 2, k = 1 $ , for which the product is: $ a_{2} \cdot a_{3} = 2 $ which is a prime number. 5. $ i = 2, k = 2 $ , for which the product is: $ a_{2} \cdot a_{3} \cdot a_{4} = 2 $ which is a prime number. In the sixth example test case there are no pairs that satisfy the conditions.

题意翻译

给定长为 $n$ 的正整数序列 $a$ 和一个正整数 $e$，求满足以下规定的有序数对 $(i,k)$ 的数量： - $i,k\ge 1$ - $i+e\times k\le n$ - $\prod\limits_{j=0}^ka_{i+e\times j}$ 是质数 多组数据，$1\le t\le 10^4,1\le e\le n\le 2\times 10^5,1\le a_i\le 10^6,1\le \sum n\le 2\times 10^5$。