Is there a particular formula for
$$ \sum_{i=1}^n i^2 \left \lfloor \frac{x}{i^2} \right \rfloor $$ where $x$ is an integer such that $n^2\leq x\leq (n+1)^2$. I have been trying for a long time and still not have reached to any conclusion.
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1I highly doubt a simple formula exists. Look at these similar questions to see the difficulty of seemingly innocuous sums like this. Would an asymptotic formula be good enough for your purposes? – Chris Grossack Apr 09 '20 at 03:57