CF1609C.Complex Market Analysis

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.

威廉在进行复杂的市场分析时遇到了如下问题：

给定一个长度为 $n$ 的数组 $a$ 和一个正整数 $e$，请计算满足以下条件的正整数对 $(i, k)$ 的个数：

• $1 \le i, k$；
• $i + e \cdot k \le n$；
• 乘积 $a_i \cdot a_{i + e} \cdot a_{i + 2 \cdot e} \cdot \ldots \cdot a_{i + k \cdot e}$ 是一个质数。

质数（或素数）是指大于 1 的自然数，且不能表示为两个更小的自然数的乘积。

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 $2 \cdot 10^5$.

每个测试包含多个测试用例。第一行包含测试用例的数量 $t$（$1 \le t \le 10\,000$）。随后是各测试用例的描述。

每个测试用例的第一行包含两个整数 $n$ 和 $e$（$1 \le e \le n \le 2 \cdot 10^5$），分别表示数组中的元素个数和数字 $e$。

第二行包含 $n$ 个整数 $a_1, a_2, \dots, a_n$（$1 \le a_i \le 10^6$），表示数组的内容。

保证所有测试用例的 $n$ 之和不超过 $2 \cdot 10^5$。

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.

对于每个测试用例，按以下格式输出答案：

输出一行，包含满足条件的数对 $(i, k)$ 的个数。

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.

在第一个样例测试用例中，有两对 $(i, k)$ 满足条件：

1. $i = 2, k = 1$，对应乘积为：$a_{2} \cdot a_{5} = 2$，这是一个质数。
2. $i = 3, k = 1$，对应乘积为：$a_{3} \cdot a_{6} = 19$，这是一个质数。

在第二个样例测试用例中，不存在满足条件的数对。

在第三个样例测试用例中，有四对 $(i, k)$ 满足条件：

1. $i = 1, k = 1$，对应乘积为：$a_{1} \cdot a_{4} = 2$，这是一个质数。
2. $i = 1, k = 2$，对应乘积为：$a_{1} \cdot a_{4} \cdot a_{7} = 2$，这是一个质数。
3. $i = 3, k = 1$，对应乘积为：$a_{3} \cdot a_{6} = 2$，这是一个质数。
4. $i = 6, k = 1$，对应乘积为：$a_{6} \cdot a_{9} = 2$，这是一个质数。

在第四个样例测试用例中，不存在满足条件的数对。

在第五个样例测试用例中，有五对 $(i, k)$ 满足条件：

1. $i = 1, k = 1$，对应乘积为：$a_{1} \cdot a_{2} = 2$，这是一个质数。
2. $i = 1, k = 2$，对应乘积为：$a_{1} \cdot a_{2} \cdot a_{3} = 2$，这是一个质数。
3. $i = 1, k = 3$，对应乘积为：$a_{1} \cdot a_{2} \cdot a_{3} \cdot a_{4} = 2$，这是一个质数。
4. $i = 2, k = 1$，对应乘积为：$a_{2} \cdot a_{3} = 2$，这是一个质数。
5. $i = 2, k = 2$，对应乘积为：$a_{2} \cdot a_{3} \cdot a_{4} = 2$，这是一个质数。

在第六个样例测试用例中，不存在满足条件的数对。