One and a half seems interesting

Let x = 2/3
Then x+x^2+x^3+... = 2

The Maclaurin series for (1 − x)⁻¹ is the geometric series

And we can see that in the limit we 2 as I have removed the first term, 1. If we add more samplers, each new sampler being 2/3 slower than the previous we still approach the Shannon-Nyquist limit of sampling at twice the arrival rate.  (I use queuing terminology instead of bandwidth).  Anyway, adding more and slow samplers we can always add enough to prevent any sampler from queuing up. It would be inefficient, but I would think this effect would show up in proofs on sampling theory.

How does the spacing between arrivals look? For each slower sample it goes as the inverse, 3/2.  Now it gets interesting. Consider packing a sphere but we allow volume spaces for the density to adjust continually. So we  make room for (3/2)^n, n=1...Max spaces inside the sphere, approaching the Shannon sampling rate. How many samplers can we put in the sphere at a maximum before we reach the sphere volume:

Here I have it.  The blue line is accumulated empty space taken in powers of 3/2. The red line is sphere volume, the X axis is radius in integer increments.  We run out of room for empty spaces at about 17.1, which is one and a half short of the ratio of the proton to electron mass: 3/2^(18.53) =1836.I guess the volume of the proton is 3/2 times the number of empty spaces allowing room for the things in motion.

This shows up on my spectral chart. This is also what makes the atom seem to be an 17 bit computer.