Tell me more ×  A famous inequality that has been proved by J.B.
Rosser and L. Schoenfeld says that  $\frac{n}{\ln n-1/2}$ <
$\pi(n)$<$\frac{n}{\ln n-3/2} , n\ge 67$.  Using this inequality
we can prove that when $\pi(n)$ divides $n$ (and this happens
infinitely often) then  $\pi(n)=\frac{n}{[\ln n-1/2]}$ (for
$n\ge 67$)  (By $[\ln n-1/2]$ we denote the integer part of $\ln
n-1/2$).  This is an exact formula for $\pi(n)$ that occurs
infinitely often.  The question is: Do we have any knowledge
about when $\pi(n)$ divides $n$ or anything in this direction?
Thank you for viewing.  share|improve this question What does it
mean to have an exact formula that holds only infinitely often,
as opposed to, well, all the time? –  Captain Oates Aug 20 at
16:45 The formula does not hold for every $n\ge 67$ but when
$\pi(n)$ divides n and also $n\ge 67$. I intended just to give
an easy and precise formula (without using ALL the previous
primes nor the sigma notation) that occurs infinitely often.
thanks for the responce th