Sure, let's write a test case generator for the given problem statement. Here is the generator function:

```python
import random

def tcgen():
    # Generate a random value for N within the constraints
    N = random.randint(1, 10**7)
    
    return N

# Example usage
print(tcgen())
```

This generator function will produce a random test case for the given problem where \(N\) is a single integer within the range \([1, 10^7]\).

Since the problem requires only a single integer \(N\) as input, the generator function is straightforward. You can call this function to generate random test cases for testing your solution to the problem.