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 \(x\).