A14738 | Complex Market Analysis
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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.
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 $2 \cdot 10^5$ .
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.
Output one line containing the number of pairs of numbers $(i, k)$ which satisfy the conditions.
输入输出样例
输入 #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 #1
2 0 4 0 5 0
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.
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.
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评测结果：Accepted