Sure! Here's a Python script that uses the Sieve of Eratosthenes algorithm to find and output prime numbers within a given range efficiently:

```python
def sieve_of_eratosthenes(lower_limit, upper_limit):
    sieve = [True] * (upper_limit + 1)
    sieve[0] = sieve[1] = False

    for current_number in range(2, int(upper_limit**0.5) + 1):
        if sieve[current_number]:
            for multiple in range(current_number * current_number, upper_limit + 1, current_number):
                sieve[multiple] = False

    prime_numbers = [num for num in range(max(2, lower_limit), upper_limit + 1) if sieve[num]]
    return prime_numbers


def main():
    lower_limit = int(input("Enter the lower limit of the range: "))
    upper_limit = int(input("Enter the upper limit of the range: "))

    prime_numbers = sieve_of_eratosthenes(lower_limit, upper_limit)

    print(f"Prime numbers in the range {lower_limit} to {upper_limit}:")
    print(prime_numbers)


if __name__ == "__main__":
    main()
```
This script takes a range of numbers as input and outputs all prime numbers within that range using the Sieve of Eratosthenes algorithm, which is efficient and suitable for handling large numbers. Just run the script, enter the lower and upper limits of the range, and it will print the prime numbers within that range.
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