# 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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