We define $f(x)$ = minimum number of positive perfect cubes that sum up to $x$.

For example,

$f(2) = 2 (2 = 1^3 + 1^3)$

$f(9) = 2 (9 = 1^3 + 2^3)$

$f(17) = 3 (17 = 1^3 + 2^3 + 2^3)$

Find $\sum_{n=1}^{10^6} f(n)$.