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 Ruby (judge verdict: Accepted)

### constant

MAX_N = 300000

### subroutines

def msnum_generator(max_n)
  yield 6
  k = 7
  loop do
    n = k + 1
    break if n > max_n
    yield n

    n = k + 6
    break if n > max_n
    yield n

    k += 7
  end
end

def gen_msprimes(max_p, msnums)
  msprimes = (max_p + 1).times.map{false}
  msnums.each{|msn| msprimes[msn] = true}

  msps = []

  for p in msnums
    if msprimes[p]
      msps << p
      for q in msnums
        pq = p * q
        break if pq > max_p
        msprimes[pq] = false
      end
    end
  end

  msps
end

### main

msnums = []
msnum_generator(MAX_N){|msn| msnums << msn}
#p msnums

msps = gen_msprimes(MAX_N, msnums)

loop do
  n = gets.to_i
  break if n == 1

  msds = []

  for msp in msps
    if n % msp == 0
      n17 = (n / msp) % 7
      if n17 == 1 || n17 == 6
        msds << msp
      end
    end
  end

  puts "#{n}: " + msds.join(' ')
end