Judging from some computer calculations $\lim_{n \rightarrow \infty} n/|\{(a,b):2^a \cdot 3^b\}| \rightarrow \infty$.  This is the GAP code I used below; you can modify the initial value of n to quite a large value and it will still work.

    n:=100000;;
    count:=0;;
    for a in [0..LogInt(n,2)] do
      for b in [0..LogInt(n,3)] do
        if(2^a*3^b<=n) then count:=count+1;;
      od;
    od;
    Print(Int(n/count),"\n");

Edit:  using the above code I searched Sloane's website and uncovered [this][1].  Specifically, these are called *3-smooth* numbers.  There's formulae there too.


  [1]: http://www.research.att.com/~njas/sequences/A071521