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
  =   x2 x3 x5 n i j k min
    .   tbl$(h,!arg)        { This creates an array. Arrays are always global in Bracmat. }
      & 1:?(0$h)
      & 2:?x2
      & 3:?x3
      & 5:?x5
      & 0:?n:?i:?j:?k
      &   whl
        ' ( !n+1:<!arg:?n
          & !x2:?min
          & (!x3:<!min:?min|)
          & (!x5:<!min:?min|)
          & !min:?(!n$h)               { !n is index into array h }
          & (   !x2:!min
              & 2*!((1+!i:?i)$h):?x2
            |
            )
          & (   !x3:!min
              & 3*!((1+!j:?j)$h):?x3
            |
            )
          & (   !x5:!min
              & 5*!((1+!k:?k)$h):?x5
            |
            )
          )
      & !((!arg+-1)$h) (tbl$(h,0)&)    { We delete the array by setting its size to 0 }
  )
& 0:?I
& whl'(!I+1:~>20:?I&put$(hamming$!I " "))
& out$
& out$(hamming$1691)
& out$(hamming$1000000)
);