A Kleiber-like growth law for prime number density between n-gon sequence terms

What follows is speculative conjecture that a Kleiber-like 5 sphere SA/V [(8/3)π²r⁴/(8/15)π²r⁵] ratio forms the limiting growth law of prime number density between consecutive terms of the sequences of n-gon quadratic sequences.

The polygon (n-gon) numbers form an infinite set of quadratic sequences with gaps between their consecutive members that span increasingly more prime numbers.

From the n=3,4,5 & 6 polygon sequences:

we can infer that the n-gon (quadratic 2D) generator for the r-th term of a n-gon sequence is:

The first five hundred odd terms of these polynomial sequences looks like the following,

n-gon sequence of degree n=3 to n=8

What we are interested is the number of prime numbers that exist between consecutive numbers in each of these sequences. This is easily accomplished in a spreadsheet. Here is a snapshot of the google sheet used to perform the count.

There are 56 primes between the 278th square number (77284) and the 279th square number (77841)

The spreadsheet delivers the following plots whose power series lines-of-best-fit are revealed to involve a 4/5 ths power law:

Between the 278th and 279th square numbers there are 56 primes

That is, the growth in the number of prime numbers that exist between consecutive numbers of an n-gon sequence is best fitted by the following 4/5th power law:

The 4/5th power is maintained even if we split the primes according to Fermat’s 4X divisibility splitting criteria:

Splitting the Primes according to Fermat 4x multipler law, and 
the ⅘ power is maintained with coefficient halved

In this case with half the primes available the multiplier coefficient halved.

As @icecolbeveridge suggested it is interesting to compare this power law to the expected Riemann counting function growth factor, 

We see that in the short run (at least) this delivers the equivalent of a 19/25 power growth -which is just shy of the required ⅘ th power (see black curve in chart below):

Riemann’s counting function, for comparison with sequence growth curves

The 4/5th nature of the power law with the 1/5th multiplicative factor has echoes of Kleiber’s law of biological animalia metabolism. Looking at the ratio of Surface Area/Volume [SA/V] of hyperspheres we note the 4/5 power and 5 pre-factor:

Surface Area and Volume of hyper-spheres with the 5-d case highlighted

We note parenthetically, with respect to the 5-dimensional nature of the spheres that this is the optimal dimension for a unit sphere:

The first five Hyperspheres Areas and Volumes

that 5-dimensional unit sphere’s have maximal volume amongst all possible dimensions while surface area is maximised at 7-dimensions:

Unit hyper-ball Volumes and Surface area versus dimension.

What then is suggestive of the preceding cursory analysis is that the following conjecture be verified by some deeper data mining:

The limiting growth function of the number density of primes between consecutive terms of polygon n-gon sequences is the Kleiber-like law being the ratio of surface area/volume of a 5 sphere.

A log-log plot of number density versus the r-th term reveals the limiting envelope (a 2-sphere being the limiting polygon):

Triangle [red] to Octagon [black] prime number density growth sequences with limiting hyper-sphere growth [purple]

This analysis extended to the 15-gon (Pentadecagon) n-gon sequences still suggests some accord with the conjectured limiting Kleiber-like 5-sphere ⅘ power law envelope:

log-log plot show 4/5th power law of prime number density growth functions for 3 to 15-gon sequences

A further conjecture suggests itself if one considers the cubic (as opposed to these 2-d quadratic) sequences generated by the Tetrahedral [cumulative Triangle], Tesserat, Pentahedral,..etc ; a.k.a the (cubic) polyhedral sequences:

The limiting growth function of prime number densities between polyhedra n-sequences is of 5/6 power as determined by the SA/V ratio of a six dimensional hypersphere.

The spreadsheet used to implement the count is in the link below. In order to move from this rampant speculation to firmer assertions a greater number of terms needs to be generated.

spreadsheet link