Sums of floors of square roots

Number Theory Level 3

Find the largest possible $n$ such that $\big\lfloor \sqrt{1} \big\rfloor + \big\lfloor \sqrt{2} \big\rfloor + \big\lfloor \sqrt{3} \big\rfloor + \cdots + \big\lfloor \sqrt{n} \big\rfloor$ is a prime number.

Clarification: $\lfloor x \rfloor$ returns the largest integer less than or equal to $x$ .