AN EXACT FORMULA TO CALCULATE 
                   THE NUMBER OF PRIMES LESS THAN OR EQUAL TO X
                  
                  


Formula:
  If x is an integer >= 4, then 

                       x  
   _____             ----  | S(k) |         
   |   |             \     | ____ | 
   |   | (x) = -1 +  /     |   k  |
   |   |             ----  --    --  
                      k=2
 
where S(k) is the Smarandache Function:  the smallest integer such that S(k)!
is divisible by k, and  |       |
                        |   a   |
                        |       |
                        --     --

means the interior integer part of a (the smallest integer greater than or
equal to a).


Proof:
  Knowing the Smarandache Function has the property that if p > 4 then
S(p) = p if only if p is prime, 
and S(k) <= k for any k,
and S(4) = 4 (the only exception from the first rule),
we easily find an exact formula for the number of primes 
less than or equal to x.


Reference:
 Seagull, L., "The smarandache Function and the number of primes up to x",
   <Mathematical Spectrum>, University of Shielfield, Vol. 28, No. 3, 1995/6,
   p. 53.