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For ease if writing I’ll denote f(m) to be sum(k=0 to m) floor(sqrt(k)) and g(k) to be floor(sqrt(k)), and finally i for floor(sqrt(k)).
Start by considering what outputs g gets when putting in various k. We notice that g(k)=p for p^2, p^2 +1,…,p^2 +2p, and that there are 2p+1 of these numbers.
Next we calculate f(p^2), which ends up being p + sum(k=1 to p-1) (p-k)(2p-2k+1). What about f(p^2 +1)? We notice a pattern for f(p^2 +q) where q is between 0 and 2p;
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