Floor Primes

For how many positive integers \(n\) are there exactly \(\lfloor \frac{n}{2} \rfloor\) or \(\lceil \frac{n}{2} \rceil\) primes less than or equal to \(n\)?

Details and assumptions

Greatest Integer Function / Floor Function: \(\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}\) refers to the greatest integer less than or equal to \(x\). For example \(\lfloor 2.3 \rfloor = 2\) and \(\lfloor -5 \rfloor = -5\).
Ceiling Function: \( \lceil x \rceil : \mathbb{R} \rightarrow \mathbb{Z} \) refers to the smallest integer that is greater than or equal to to \(x\). For example, \( \lceil 2.3 \rceil = 3 \) and \( \lceil -5 \rceil = -5 \).

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