Formula
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AN EXACT FORMULA TO CALCULATE THE NUMBER OF PRIMES LESS THAN OR EQUAL TO X
Formula: If x is an integer >= 4, then _____ | | | | (x) = -1 + | |
x ---\ / ---k=2
| S(k) | | ____ | | k | ---
where S(k) is the Smarandache Function: is divisible by k, and | | | a | | | ---
the smallest integer such that S(k)!
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.
Formula: If x is an integer >= 4, then _____ | | | | (x) = -1 + | |
x ---\ / ---k=2
| S(k) | | ____ | | k | ---
where S(k) is the Smarandache Function: is divisible by k, and | | | a | | | ---
the smallest integer such that S(k)!
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.
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