number of integer solutions for $2^x\cdot 3^y$ Can someone help me figure out how to calculate the number of integer solutions to the equation: $N\ge2^x3^y|x,y\in\Bbb Z+$ 
For example if $N$ is $30$ we have $2\cdot 3$, $2^2\cdot 3$, $2^3\cdot 3$, $2\cdot 3^2$ so $4$ solutions
I know the number of integer solutions to $N\ge2^x$ is just $\left \lfloor \ln{N}/\ln{2} \right \rfloor $
So I was thinking something along the lines of taking the sum of $\log_2\left(\frac{N}{3^i}\right)$ from $i=1..(\log_3(N))$
Any help is greatly appreciated
 A: 
The problem $2^x 3^y \le 30$ can be solved graphically as follows.
Taking the logarithm of both sides and simplifying, we get
\begin{align}
   2^x 3^y &\le 30 \\
   (\ln 2)x + (\ln 3)y &\le \ln 30 \\
\end{align}
The line above is the line described by $(\ln 2)x + (\ln 3)y = \ln 30$.
The indicated lattice points are the $(x,y)$ solutions to $ 2^x 3^y \le 30$
A: You want to count all the points with positive integer coordinates which lie on or below the line
$$ y=-x\log_3(2)+N\log_3(N)\tag{1}$$
which one obtains by taking $\log_3$ of both sides of 
$$ 2^x3^y=N $$
and solving for $y$.
A formula for computing the number $C(N)$ of points having positive integer coordinates and lying on or below the line given by $(1)$ is
$$ C(N)=\sum_{k=1}^{\left\lfloor\log_2(N/3)\right\rfloor}\left\lfloor\log_3\left(\frac{N}{2^k}\right)\right\rfloor\tag{2} $$
For example, for $N=243$ this gives
$$ C(243)=\sum_{k=1}^{6}\left\lfloor\log_3\left(\frac{243}{2^k}\right)\right\rfloor=4+3+3+2+1+1=14 $$
Here is a GeoGebra link with a slider where you can change the value of $N$.
Below is a static illustration.

