Predicting the number of primes less than is a contentious mathematical subject. Here are some of my thoughts, from messing around with factorials and the natural log. Note that many of my arguments are not rigorous. Feel free to give feedback below.
Prime-factor all of the terms on the right:
where is the "-adic valuation of " (exponent of in the prime factorization of ).
Take the natural logarithm of both sides:
Recall Stirling's asymptotic formula for the factorial:
All the other terms on the right hand are small compared to when is very large, so
So far, we have managed to link the natural logarithm with the number of primes in the prime factorization of . Let's focus on . One of Legendre's many theorems (1808) tells us that
Now, Here's where the guessing comes in: as , I conjecture that , and in turn that . I will assume from now on that this is true for large . As such,
We thus have
(green points) v.s. (red line) from to . Created using Desmos