<s>[INST] The asymptotic of the number of integers that are sums of three nonnegative cubes Let $c(n) $ be the number of distinct integers between $0 $ and $n $ of the form $ a^3 + b^3 + c^3$, meaning the sum of $3$ nonnegative cubes. 
$C(n) = O( n \space \ln(n)^x ) $
Find and prove the optimal real value of $x$.
 [/INST] Maybe this could help:
The amount of cubes lower than or equal to $n$ is $\lfloor\sqrt[3]{n}\rfloor$.
If we just have $A^3+B^3$ instead of $A^3+B^3+C^3$ we get:
$$C(n)=\dfrac{1}{2}\sum_{i=1}^{\lfloor\sqrt[3]{n-1}\rfloor}\lfloor\sqrt[3]{n-i^3}\rfloor$$
And for $A^3+B^3+C^3$ (which is the result you want):
$$C(n)=\dfrac{1}{6}\sum_{i=1}^{\lfloor\sqrt[3]{n-2}\rfloor}\sum_{j=1}^{\lfloor\sqrt[3]{n-i^3-1}\rfloor}\lfloor\sqrt[3]{n-i^3-j^3}\rfloor$$
This formula basically  just checks whether there exists a $C$ for every combination of $A$ and $B$.
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