Number of primes <= N - - - - my challenge problem

lookagain

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For \(\displaystyle 10 \le N \le 3,000, \ \ \) can you search for/determine a formula that approximates
the number of prime numbers less than or equal to the integer N that lies within that specified range?

You can't use \(\displaystyle \ \dfrac{N}{ \ ln(N) - 1 \ }, \ \ \ where \ \ ln(N) \ \ means \ \ log_e(N).\)

Your approximation formula should give results that are "comparable" to the one immediately above, though.

The Internet search is encouraged for this.


\(\displaystyle \pi(N) \)= the exact number of prime numbers less than or equal to N


Examples:

\(\displaystyle \pi(10) = 4\)

\(\displaystyle \pi(100) = 25\)

\(\displaystyle \pi(1,000) = 168\)



(I have my own "best" formula picked out, but it is not to be revealed at the outset of this thread.)
 
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