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).


SetBatchLines, -1
Msgbox % hamming(1,20)
Msgbox % hamming(1690)
return
 
hamming(first,last=0)
{
	if (first < 1)
		ans=ERROR
 
	if (last = 0)
		last := first
 
	i:=0, j:=0, k:=0
 
	num1 := ceil((last * 20)**(1/3))
	num2 := ceil(num1 * ln(2)/ln(3))
	num3 := ceil(num1 * ln(2)/ln(5))
 
	loop
	{
		H := (2**i) * (3**j) * (5**k)
		if (H > 0)
			ans = %H%`n%ans%
		i++
		if (i > num1)
		{
			i=0
			j++
			if (j > num2)
			{
				j=0
				k++
			}
		}
		if (k > num3)
			break
	}
	Sort ans, N
 
	Loop, parse, ans, `n, `r 
	{
		if (A_index > last)
			break
		if (A_index < first)
			continue
		Output = %Output%`n%A_LoopField%
	}
 
	return Output
}