Certainly! Below is a test case generator for the problem statement that counts the number of Shichi-Go-San numbers between 1 and N.

### Test Case Generator

```python
import random

def tcgen():
    N = random.randint(1, 10**9)
    return N

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

### Explanation of the Generator

1. **Random Integer Generation**: The generator will produce a single integer \( N \) within the specified range \( 1 \leq N < 10^9 \).
2. **Output**: The function returns this integer \( N \) which then can be used as an input to the solution function.

### Example Outputs from the Generator

Here are a few example outputs that the generator might produce:

- 52346789
- 1000000
- 987654321

You can use these generated test cases to check the correctness and performance of your solution for the Shichi-Go-San numbers problem.

### Python Script to Run the Generator

```python
import random

def tcgen():
    N = random.randint(1, 10**9)
    return N

if __name__ == "__main__":
    # Generate and print a test case
    test_case = tcgen()
    print(test_case)
```

This script will generate a random test case each time it's run, which can help in thorough testing of the solution to ensure it works for a wide range of inputs.