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).


(defun next-hamm (factors seqs)
  (let ((x (apply #'min (map 'list #'first seqs))))
    (loop for s in seqs
	  for f in factors
	  for i from 0
	  with add = t do
	  (if (= x (first s)) (pop s))
	  ;; prevent a value from being added to multiple lists
	  (when add
	    (setf (elt seqs i) (nconc s (list (* x f))))
	    (if (zerop (mod x f)) (setf add nil)))
    finally (return x))))
 
(loop with factors = '(2 3 5)
      with seqs    = (loop for i in factors collect '(1))
      for n from 1 to 1000001 do
      (let ((x (next-hamm factors seqs)))
	(if (or (< n 21)
		(= n 1691)
		(= n 1000000)) (format t "~d: ~d~%" n x))))