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We define $f(x) =$ minimum number of positive perfect cubes that sum up to $x$.
For example,$$f(17) = 3(17 = 1^3 + 2^3 + 2^3)$$ $$f(9) = 2 (9 = 1^3 + 2^3)$$ How can we find $f(x)$ for any number $x$?

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    $\begingroup$ Are you familiar with the Waring Problem? An exact solution isn't known for all $x$, but good upper bounds are known. $\endgroup$ – Steven Stadnicki Jun 30 '17 at 18:41
  • $\begingroup$ @DietrichBurde Not a duplicate. That was about $\max_x f(x)$. Here we want to compute $f(x)$ for any given $x$. $\endgroup$ – Robert Israel Jun 30 '17 at 18:52
  • $\begingroup$ @RobertIsrael Oh, sorry, this was a mistake. Still, this page is relevant. For example, $f(23)=9$. $\endgroup$ – Dietrich Burde Jun 30 '17 at 18:52
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    $\begingroup$ You might look at OEIS sequence A002376 and links there. $\endgroup$ – Robert Israel Jun 30 '17 at 19:04
  • $\begingroup$ $G(3)=9$ so the $ 1 \leq f(x) \leq 9$. $\endgroup$ – Ahmad Jun 30 '17 at 19:10

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