Sums of floors of square roots

Find the largest possible nn such that 1+2+3++n\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: x\lfloor x \rfloor returns the largest integer less than or equal to xx.

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