Problem:
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

Solution in C++ (judge verdict: Accepted)

#include <algorithm>
#include <iostream>
#include <vector>

using namespace std;

typedef vector<int> VI;
typedef vector<bool> VB;

VI ms_prime_factor ( void )
{
	const int SIZE = 300010;
	VB is_ms_prime_number( SIZE, false );
	VI res;
	for ( int i = 0; 7 * i + 10 < SIZE; ++i ) {
		is_ms_prime_number[7 * i + 1] = true;
		is_ms_prime_number[7 * i + 6] = true;
	}

	for ( int i = 2; i < SIZE; ++i ) {
		if ( !is_ms_prime_number[i] ) {
			continue;
		}
		for ( int j = i + i; j < SIZE; j += i ) {
			is_ms_prime_number[j] = false;
		}
		res.push_back( i );
	}
	return res;
}

int main ( void )
{
	VI p_factor = ms_prime_factor();
	int num;
	while ( cin >> num, num != 1 ) {
		cout << num << ':';
		for ( int i = 0; i < p_factor.size(); ++i ) {
			if ( num % p_factor[i] == 0 ) {
				cout << ' ' << p_factor[i];
			}
		}
		cout << endl;
	}
	return 0;
}