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).


module ISet = Set.Make(struct type t = int let compare=compare end)
 
let pq = ref (ISet.singleton 1)
 
let next () =
  let m = ISet.min_elt !pq in
  pq := ISet.(remove m !pq  |> add (2*m) |> add (3*m) |> add (5*m));
  m
 
let () =
 
  print_string "The first 20 are: ";
 
  for i = 1 to 20
  do
    Printf.printf "%d " (next ())
  done;
 
  for i = 21 to 1690
  do
    ignore (next ())
  done;
 
  Printf.printf "\nThe 1691st is %d\n" (next ());