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).


 
# syntax: gawk -M -f hamming_numbers.awk
BEGIN {
    for (i=1; i<=20; i++) {
      printf("%d ",hamming(i))
    }
    printf("\n1691: %d\n",hamming(1691))
    printf("\n1000000: %d\n",hamming(1000000))
    exit(0)
}
function hamming(limit,  h,i,j,k,n,x2,x3,x5) {
    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])
}
function min(x,y) {
    return((x < y) ? x : y)
}
 