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).


FUNCTION min (a, b)
    IF a < b THEN min = a ELSE min = b
END FUNCTION
 
FUNCTION Hamming (limit)
    DIM h(limit)
 
    h(0) = 1
    x2 = 2
    x3 = 3
    x5 = 5
    i = 0
    j = 0
    k = 0
    FOR n = 1 TO limit
        h(n) = min(x2, min(x3, x5))
        IF x2 = h(n) THEN
            i = i + 1
            x2 = 2 * h(i)
        END IF
        IF x3 = h(n) THEN
            j = j + 1
            x3 = 3 * h(j)
        END IF
        IF x5 = h(n) THEN
            k = k + 1
            x5 = 5 * h(k)
        END IF
    NEXT n
    Hamming = h(limit - 1)
END FUNCTION
 
PRINT "The first 20 Hamming numbers are :"
FOR i = 1 TO 20
    PRINT Hamming(i); " ";
NEXT i
 
PRINT
PRINT "H( 1691) = "; Hamming(1691)