Hi Everyone,

I am a math amateur who for the past year has been working on better understanding Bertrand's Postulate, the Ramanujan Primes, and the recent expansion of Bertrand's Postulate (always a prime between 2x and 3x and always a prime between 3x and 4x) using elementary methods.

I've been working with least prime factors and primorials and I came up with a counting formula that I have not seen elsewhere. It is quite similar to the standard prime counting formula using floor functions and it is using elementary methods so it is most likely uninteresting. I hope you don't mind me posting a sketch of it here.

I am presenting it here in hopes that experts can steer me to more modern analytic methods that accomplish the same thing in a better way. I would also be interested in understanding why the new methods are superior to the elementary methods.

The counting formula accomplishes the following:

Let $p_k$ be any prime. The formula provides an exact count of the number of least prime factors greater than $p_k$ in the range $r_{start}$ (exclusive) and $r_{end}$ (inclusive).

The formula consists of $2^{k-2}$ subformulas where each subformula looks something like this:

Least Prime Factor (5 or greater) between $x_{start}$ and $x_{end}$ =

$2\lfloor\frac{x_{end}}{6}\rfloor + \lfloor\frac{(x_{end} \% 6) + 3}{4}\rfloor - 2\lfloor\frac{x_{start}}{6}\rfloor - \lfloor\frac{(x_{start} \% 6) + 3}{4}\rfloor$

where $x_{end} \% 6$ is the value congruent to $x$ modulo $6$.

Note: The above formula, for example, is the expression for finding the number of least prime factors greater than $3$ in the range $r_{start}$ to $r_{end}$.

To give another example, if I wanted to count the number of least prime factors greater than $p_{6} = 13$, then the formula consists of $2^{6-2} = 16$ subformulas where each subformula is roughly similar to the example above.

Thanks very much.