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).


' FB 1.05.0 Win64
 
' The biggest integer which FB supports natively is 8 bytes so unable
' to calculate 1 millionth Hamming number without using an external 
' "bigint" library such as GMP
 
Function min(x As Integer, y As Integer) As Integer
  Return IIf(x < y, x, y)
End Function
 
Function hamming(n As Integer) As Integer
  Dim h(1 To n) As Integer
  h(1) = 1
  Dim As Integer i = 1, j = 1, k = 1   
  Dim As Integer x2 = 2, x3 = 3, x5 = 5
 
  For m As Integer = 2 To n
    h(m) = min(x2, min(x3, x5))
    If h(m) = x2 Then 
      i += 1
      x2 = 2 * h(i)
    End If
    If h(m) = x3 Then
      j += 1
      x3 = 3 * h(j)
    End if
    If h(m) = x5 Then
      k += 1
      x5 = 5 * h(k)
    End If
  Next     
 
  Return h(n)
End Function
 
Print "The first 20 Hamming numbers are :"
For i As Integer = 1 To 20
  Print hamming(i); " ";
Next
Print : Print
Print "The 1691st hamming number is :"
Print hamming(1691) 
Print
Print "Press any key to quit"
Sleep