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 JAVA (judge verdict: Accepted)

import java.util.Arrays;
import java.util.PriorityQueue;
import java.util.Scanner;

//Monday-Saturday Prime Factors
public class Main{

	public static void main(String[] args) {
		Scanner sc = new Scanner(System.in);
		int[] msf = new int[85713];
		int id = 0;
		msf[id++] = 6;
		for(int i=7;i<300000;i+=7){
			if(i+1<300000){
				msf[id++] = i+1;
			}
			if(i+6<300000){
				msf[id++] = i+6;
			}
		}
		boolean[] mspf = new boolean[300000];
		Arrays.fill(mspf, true);
		for(int i=0;i<85713;i++){
			int x = msf[i];
			if(mspf[x]){
				for(int j=2*x;j<300000;j+=x){
					mspf[j] = false;
				}
			}
		}
		while(true){
			int x = sc.nextInt();
			if(x==1)break;
			System.out.print(x + ":");
			PriorityQueue<Integer> q = new PriorityQueue<Integer>();
			for(int i=0;i<85713;i++){
				if(x < msf[i])break;
				if(mspf[msf[i]] && x%msf[i]==0){
					if(!q.contains(msf[i]))q.add(msf[i]);
				}
			}
			while(!q.isEmpty())System.out.print(" " + q.poll());
			System.out.println();
		}
	}
}