As you know, I have been working with the R-sequence for some time. Some time ago I found this interesting feature that the R-sequence, part of it (3,9,10,27,28,30,81) resembles https://oeis.org/A060140 and this means, by its definition, the numbers of the form 9x+1 that occupy the same positions in S that 1 occupies in the infinite Fibbonaci word (https://oeis.org/A003849). How interesting this is that the prime numbers might be closely connected to the Fibonacci sequence?
To grasp the more general pick at the universe of the numbers, we’d have to understand more about the Fibonacci sequence and the rationale for its potential connection with the primes. Still, wanted to share that.
Btw, my new task:
For all integers find all , , such that , where are consecutive primes, for positive integers .
Below R-sequence sketched in R.
Below, in blue, you can see the x/lgx from the PNT together with the R-sequence. The amount of the elements in the R-sequence is the same as the one of primes, thus the idea to approx. the PNT theorem with the use of the R-sequence.
Below, started to look for the very approximation of the PNT with the use of the R-sequence. The thing is that we could see a cunning resemblance between the number of primes and the R-sequence. And R-sequence itself is more embedded into the life of numbers than the infamous PNT. Below a part of research I conducted where I tried to use a linear approximation instead of a logarithmical one. That could work only if we were summing over a number of such approximations, knowing exactly how to do it.
Below, you can see how the R-sequence looks in comparison to the PNT formula.