I want to plot a surface in 3D space; the cartesian product
$Z=\mathbb{N}_2^{\times}\times\langle2\rangle$
Where
$\mathbb{N}_2^{\times}=\{1,3,5,7,9,\ldots\}$ and
$\langle2\rangle=\{1,2,4,8,16,\ldots\}$
$x\in \mathbb{N}_2^{\times}$ on the x-co-ordinate, $y\in\langle2\rangle$ on the y-co-ordinate and their product on the z-co-ordinate. So this will show the positive integers as a hyperbolic paraboloid which gives a visual ordering of the integers $z=xy$ by their 2-adic metric $1/x$.
Then I want to graph the orbits of the function $z_{n+1}=3z_n+y_n$ on this surface to add insight to the calculus involved in solving the Collatz conjecture.
The Collatz conjecture is equivalent to the statement that every orbit of the above successor relation on $z$ converges to the intersection of the hyperbolic paraboloid $Z$ and the plane $x=1$. My prior calculus argument was rejected by the community so I'm hoping to construct a more visual argument to help me underpin the calculus, my communication of it, and the ease with which some willing person might help me refine the argument.
I've spent a good degree of time investigating various graphing software such as Geogebra but I'm not finding an efficient way to plot this surface and the orbits. This is probably a combination of not choosing the right software and my not knowing how to use it.
Can anybody therefore suggest a suitable software for plotting the above surface and trajectories along it, and any hints to get me started?