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).


hamming = 1 : map (2*) hamming `union` map (3*) hamming `union` map (5*) hamming
 
union a@(x:xs) b@(y:ys) = case compare x y of
            LT -> x : union  xs  b
            EQ -> x : union  xs  ys
            GT -> y : union  a   ys
 
main = do
    print $ take 20 hamming
    print  (hamming !! (1691-1), hamming !! (1692-1))
    print $ hamming !! (1000000-1)
 
-- Output:
-- [1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,25,27,30,32,36]
-- (2125764000,2147483648)
-- 519312780448388736089589843750000000000000000000000000000000000000000000000000000000