Hamming numbers are numbers of the form  

    H = 2i × 3j × 5k
           where 
     i,  j,  k  ≥  0 

Hamming numbers   are also known as   ugly numbers   and also   5-smooth numbers   (numbers whose prime divisors are less or equal to 5).



Task

Generate the sequence of Hamming numbers, in increasing order.   In particular:

 Show the   first twenty   Hamming numbers.
 Show the   1691st   Hamming number (the last one below   231).
 Show the   one millionth   Hamming number (if the language – or a convenient library – supports arbitrary-precision integers).


Related tasks

 Humble numbers
 N-smooth numbers


References

 Wikipedia entry:   Hamming numbers     (this link is re-directed to   Regular number).
 Wikipedia entry:   Smooth number
 OEIS entry:   A051037   5-smooth   or   Hamming numbers
 Hamming problem from Dr. Dobb's CodeTalk (dead link as of Sep 2011; parts of the thread here and here).


to init.ham
  ; queues
  make "twos   [1]
  make "threes [1]
  make "fives  [1]
end
to next.ham
                          localmake "ham first :twos
  if less? first :threes :ham [make "ham first :threes]
  if less? first :fives  :ham [make "ham first :fives]
 
  if equal? :ham first :twos   [ignore dequeue "twos]
  if equal? :ham first :threes [ignore dequeue "threes]
  if equal? :ham first :fives  [ignore dequeue "fives]
 
  queue "twos   :ham * 2
  queue "threes :ham * 3
  queue "fives  :ham * 5
 
  output :ham
end
 
init.ham
repeat      20 [print  next.ham]
repeat 1690-20 [ignore next.ham]
print next.ham