Naturally occuring counting process with a 1/log asymptotics? Besides prime numbers, is there another physically realizable counting process that exhibits a 1/log density ? The reason I am posting this question is that we are measuring the response of a quantum tunneling based counting device and it is showing 1/log asymptotics which we have matched up to 50 million primes. The measurement results can be found here
https://arxiv.org/abs/1711.11032
In other words, is the 1/log density unique to prime number generation ? 
Any suggestions would be helpful before we set up an experiment to precisely measure the electron transport process.
 A: No. One can create many sets of integers whose density decays like 1/log. An easy variant of the set of primes is the set of integers $n$ not divisible by any prime less than $n^\alpha$, for some fixed $0<\alpha<\frac12$.
Here's another example: the set of all integers $n$ such that $n$ is divisible by the number of digits of $n$. (So all one digit numbers, all even two-digit numbers, all three-digit multiples of 3, ..., all ten-digit numbers ending in 0, etc.) The number of digits of $n$ is $\lfloor \log_{10}(n) \rfloor + 1$, so the 1/log behavior emerges naturally from the definition.
A stranger possibility is to take numbers $n$ such that both $n$ and $n+1$ can be written as the sum of two squares, $n=a^2+b^2$ and $n+1=c^2+d^2$. The set of numbers that can be written as the sum of two squares has density decaying like 1/sqrt(log), and this "twin" construction essentially multiplies those two "probabilities" together (up to the leading constant).
As a general rule: when people see that the set of primes has a particular property, they tend to overestimate how unique the set of primes is with respect to that property (or how characteristic that property is of the set of primes).
