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).


using System;
using System.Numerics;
using System.Linq;
 
namespace Hamming {
 
    class MainClass {
 
        public static BigInteger Hamming(int n) {
            BigInteger two = 2, three = 3, five = 5;
            var h = new BigInteger[n];
            h[0] = 1;
            BigInteger x2 = 2, x3 = 3, x5 = 5;
            int i = 0, j = 0, k = 0;
 
            for (int index = 1; index < n; index++) {
                h[index] = BigInteger.Min(x2, BigInteger.Min(x3, x5));
                if (h[index] == x2) x2 = two * h[++i];
                if (h[index] == x3) x3 = three * h[++j];
                if (h[index] == x5) x5 = five * h[++k];
            }
            return h[n - 1];
        }
 
        public static void Main(string[] args) {
            Console.WriteLine(string.Join(" ", Enumerable.Range(1, 20).ToList().Select(x => Hamming(x))));
            Console.WriteLine(Hamming(1691));
            Console.WriteLine(Hamming(1000000));
        }
    }
}