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).


import java.math.BigInteger;
import java.util.PriorityQueue;
 
final class Hamming {
    private static BigInteger THREE = BigInteger.valueOf(3);
    private static BigInteger FIVE = BigInteger.valueOf(5);
 
    private static void updateFrontier(BigInteger x,
                                       PriorityQueue<BigInteger> pq) {
        pq.offer(x.shiftLeft(1));
        pq.offer(x.multiply(THREE));
        pq.offer(x.multiply(FIVE));
    }
 
    public static BigInteger hamming(int n) {
        if (n <= 0)
            throw new IllegalArgumentException("Invalid parameter");
        PriorityQueue<BigInteger> frontier = new PriorityQueue<BigInteger>();
        updateFrontier(BigInteger.ONE, frontier);
        BigInteger lowest = BigInteger.ONE;
        for (int i = 1; i < n; i++) {
            lowest = frontier.poll();
            while (frontier.peek().equals(lowest))
                frontier.poll();
            updateFrontier(lowest, frontier);
        }
        return lowest;
    }
 
    public static void main(String[] args) {
        System.out.print("Hamming(1 .. 20) =");
        for (int i = 1; i < 21; i++)
             System.out.print(" " + hamming(i));
        System.out.println("\nHamming(1691) = " + hamming(1691));
        System.out.println("Hamming(1000000) = " + hamming(1000000));
    }
}