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).


n = 40;
 
powers_2 = 2.^[0:n-1];
powers_3 = 3.^[0:n-1];
powers_5 = 5.^[0:n-1];
 
matrix = powers_2' * powers_3;
powers_23 = sort(reshape(matrix,n*n,1));
 
 
matrix = powers_23 * powers_5;
powers_235 = sort(reshape(matrix,n*n*n,1));
 
%
% Remove the integer overflow values.
%
powers_235 = powers_235(powers_235 > 0);
 
disp(powers_235(1:20))
disp(powers_235(1691))