Problem p00735 - Generation 2

Orig Description

Problem B: Monday-Saturday Prime Factors
Chief Judge's log, stardate 48642.5.
We have decided to make a problem from elementary number theory.
The problem looks like finding all prime factors of a positive integer,
but it is not.
A positive integer whose remainder divided by 7 is either 1 or 6 is called
a 7N+{1,6} number.
But as it is hard to pronounce,
we shall call it a Monday-Saturday number.
For Monday-Saturday numbers a and b,
we say a is a Monday-Saturday divisor of b
if there exists a Monday-Saturday number x
such that ax = b.
It is easy to show that
for any Monday-Saturday numbers a and b,
it holds that a is
a Monday-Saturday divisor of b
if and only if
a is a divisor of b in the usual sense.
We call a Monday-Saturday number a Monday-Saturday prime
if it is greater than 1 and has no Monday-Saturday divisors
other than itself and 1.
A Monday-Saturday number which is a prime in the usual sense
is a Monday-Saturday prime
but the converse does not always hold.
For example, 27 is a Monday-Saturday prime
although it is not a prime in the usual sense.
We call a Monday-Saturday prime
which is a Monday-Saturday divisor of a Monday-Saturday number a
a Monday-Saturday prime factor of a.
For example, 27 is one of the Monday-Saturday prime factors of 216,
since 27 is a Monday-Saturday prime
and 216 = 27 × 8 holds.
Any Monday-Saturday number greater than 1
can be expressed as a product of one or more Monday-Saturday primes.
The expression is not always unique
even if differences in order are ignored.
For example,
216 = 6 × 6 × 6 = 8 × 27
holds.
Our contestants should write a program that outputs
all Monday-Saturday prime factors
of each input Monday-Saturday number.
Input
The input is a sequence of lines each of which contains a single
Monday-Saturday number.
Each Monday-Saturday number is greater than 1
and less than 300000 (three hundred thousand).
The end of the input is indicated by a line
containing a single digit 1.
Output
For each input Monday-Saturday number,
it should be printed, followed by a colon `:'
and the list of its Monday-Saturday prime factors on a single line.
Monday-Saturday prime factors should be listed in ascending order
and each should be preceded by a space.
All the Monday-Saturday prime factors should be printed only once
even if they divide the input Monday-Saturday number more than once.
Sample Input
205920
262144
262200
279936
299998
1
Output for the Sample Input
205920: 6 8 13 15 20 22 55 99
262144: 8
262200: 6 8 15 20 50 57 69 76 92 190 230 475 575 874 2185
279936: 6 8 27
299998: 299998

Extracted Specification

An unspecified number of lines, each containing a single integer \(X\) (1 < \(X\) < 300000). The end of the input is indicated by a line containing a single digit 1.

### Example Input:
```
205920
262144
262200
279936
299998
1
```

### Function Signature:
Write a function `f(inputs)` that takes in the inputs.

```python
def f(inputs: List[int]):
    '''
    inputs: a list of integers
    '''
```

Test Case Generator

Sure, I can help with that. Below is a Python function to generate test cases for the "Monday-Saturday Prime Factors" problem.

```python
import random

def is_monday_saturday_number(n):
    return n % 7 == 1 or n % 7 == 6

def generate_monday_saturday_number(limit):
    while True:
        num = random.randint(2, limit)
        if is_monday_saturday_number(num):
            return num

def tcgen(num_cases=5, limit=300000):
    cases = [generate_monday_saturday_number(limit) for _ in range(num_cases)]
    cases.append(1)  # End of input indicator
    return cases

# Example usage:
test_cases = tcgen()
for case in test_cases:
    print(case)
```

### Explanation:
1. **is_monday_saturday_number(n)**: This helper function checks if a number is a Monday-Saturday number (7N+{1,6}).
2. **generate_monday_saturday_number(limit)**: This helper function generates a random Monday-Saturday number that is greater than 1 and less than the given limit.
3. **tcgen(num_cases=5, limit=300000)**: This function generates the specified number of test cases (default is 5) and ensures all generated numbers are Monday-Saturday numbers within the given range. It also appends a `1` at the end to indicate the end of input, as specified in the problem statement.

### Example usage:
The generated test cases can be used as input to your program. The function prints each test case on a new line as required.

Feel free to adjust the `num_cases` and `limit` parameters if you need more test cases or a different range for the Monday-Saturday numbers.

Extract Arguments

def extract_arguments(fh):
    inputs = []
    while True:
        line = fh.readline().strip()
        if line == '1':
            break
        inputs.append(int(line))
    return inputs