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).


(define smerge
  [X|Xs] [Y|Ys] -> [X | (freeze (smerge (thaw Xs) [Y|Ys]))] where (< X Y)
  [X|Xs] [Y|Ys] -> [Y | (freeze (smerge [X|Xs] (thaw Ys)))] where (> X Y)
  [X|Xs] [_|Ys] -> [X | (freeze (smerge (thaw Xs) (thaw Ys)))])
 
(define smerge3
  Xs Ys Zs -> (smerge Xs (smerge Ys Zs)))
 
(define smap
  F [S|Ss] -> [(F S)|(freeze (smap F (thaw Ss)))])
 
(set hamming [1 | (freeze (smerge3 (smap (* 2) (value hamming))
                                   (smap (* 3) (value hamming))
                                   (smap (* 5) (value hamming))))])
 
(define stake
  _      0 -> []
  [S|Ss] N -> [S|(stake (thaw Ss) (1- N))])
 
(stake (value hamming) 20)