Orig Description
Prime-Factor Prime
A positive integer is called a "prime-factor prime" when the number of its prime factors is prime. For example, $12$ is a prime-factor prime because the number of prime factors of $12 = 2 \times 2 \times 3$ is $3$, which is prime. On the other hand, $210$ is not a prime-factor prime because the number of prime factors of $210 = 2 \times 3 \times 5 \times 7$ is $4$, which is a composite number.
In this problem, you are given an integer interval $[l, r]$. Your task is to write a program which counts the number of prime-factor prime numbers in the interval, i.e. the number of prime-factor prime numbers between $l$ and $r$, inclusive.
Input
The input consists of a single test case formatted as follows.
$l$ $r$
A line contains two integers $l$ and $r$ ($1 \leq l \leq r \leq 10^9$), which presents an integer interval $[l, r]$. You can assume that $0 \leq r-l < 1,000,000$.
Output
Print the number of prime-factor prime numbers in $[l,r]$.
Sample Input 1
1 9
Output for Sample Input 1
4
Sample Input 2
10 20
Output for Sample Input 2
6
Sample Input 3
575 57577
Output for Sample Input 3
36172
Sample Input 4
180 180
Output for Sample Input 4
1
Sample Input 5
9900001 10000000
Output for Sample Input 5
60997
Sample Input 6
999000001 1000000000
Output for Sample Input 6
592955
In the first example, there are 4 prime-factor primes in $[l,r]$: $4,6,8,$ and $9$.
Extracted Specification
An integer M (1 ≤ M ≤ 10^9), representing the start of an interval.
An integer N such that (M ≤ N ≤ 10^9), representing the end of an interval. You can assume that (0 ≤ N - M < 1,000,000).
### Example Input:
```
1 9
```
### Function Signature:
Write a function f(M, N) that takes in the input.
def f(M: int, N: int):
'''
M: an integer
N: an integer
'''
Test Case Generator
To create a test case generator for the "Prime-Factor Prime" problem, we need to generate valid intervals \([l, r]\) within the specified constraints. Here’s a Python function that generates such test cases:
```python
import random
def tcgen():
# Generate l and r such that 1 ≤ l ≤ r ≤ 10^9
# Also ensure that 0 ≤ r - l < 1,000,000
l = random.randint(1, 10**9 - 1)
max_r = min(l + 999999, 10**9)
r = random.randint(l, max_r)
return l, r
# Example usage
for _ in range(5):
l, r = tcgen()
print(f"{l} {r}")
```
This function ensures that the generated values for \(l\) and \(r\) meet the problem constraints:
1. \(1 ≤ l ≤ r ≤ 10^9\)
2. \(0 ≤ r - l < 1,000,000\)
### Explanation:
1. We first choose a random integer \(l\) within the range \([1, 10^9 - 1]\).
2. We then set the upper limit for \(r\) as the minimum of \(l + 999999\) and \(10^9\) to satisfy the constraint \(0 ≤ r - l < 1,000,000\).
3. Finally, we choose \(r\) as a random integer within the range \([l, max\_r]\).
This keeps the interval within the allowed limits while ensuring variability in the test cases. You can use this function to generate multiple test cases for your competitive programming needs.
Extract Arguments
def extract_arguments(fh):
l, r = map(int, fh.readline().strip().split())
return l, r