The Golomb's self-describing sequence $(G(n))$ is the only nondecreasing sequence of natural numbers such that $n$ appears exactly $G(n)$ times in the sequence. The values of $G(n)$ for the first few $n$ are

$$\begin{array}{ccccccccccccccccc}n& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15& \dots \\ G(n)& 1& 2& 2& 3& 3& 4& 4& 4& 5& 5& 5& 6& 6& 6& 6& \dots \end{array}$$

You are given that $G({10}^3)=86$, $G({10}^6)=6137$.
You are also given that $\sum G(n^3)=153506976$ for $1\le n<{10}^3$.

Find $\sum G(n^3)$ for $1\le n<{10}^6$.