4. Determine all natural numbers $n$ for which:

$$
[\sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+[\sqrt[3]{n}]=2 n
$$
Solution. Let $a_{k}=[\sqrt[3]{k}]-2$, for $k=1,2,3, \ldots$. We need to determine all natural numbers $n$ for which $a_{1}+a_{2}+\ldots+a_{n}=0$. Note that

$$
\begin{array}{ll}
a_{k}=-1, & 1 \leq k \leq 7 \\
a_{k}=0, & 8 \leq k \leq 26 \\
a_{k}=1, & 27 \leq k \leq 63 \\
a_{k} \geq 2, & k \geq 64
\end{array}
$$

Clearly, the sum $a_{1}+a_{2}+\ldots+a_{n}$ is equal to 0 if and only if $n=7+26=33$.

## II year