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).


cat hamming_numbers.bc
define min(x,y) {
 if (x < y) {
	return x
 } else {
	return y
 }
}
define hamming(limit) {
 i = 0
 j = 0
 k = 0
 h[0] = 1
 x2 = 2
 x3 = 3
 x5 = 5
 for (n=1; n<=limit; n++) {
  h[n] = min(x2,min(x3,x5))
  if (h[n] == x2) { x2 = 2 * h[++i] }
  if (h[n] == x3) { x3 = 3 * h[++j] }
  if (h[n] == x5) { x5 = 5 * h[++k] }
 }
 return (h[limit-1])
}
for (lab=1; lab<=20; lab++) {
 hamming(lab)
}
hamming(1691)
hamming(1000000)
quit
 