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Problem 884

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Removing Cubes

Starting from a positive integer $n$, at each step we subtract from $n$ the largest perfect cube not exceeding $n$, until $n$ becomes $0$.

For example, with $n=100$ the procedure ends in $4$ steps:
$$100\stackrel{-4^3}{\to }36\stackrel{-3^3}{\to }9\stackrel{-2^3}{\to }1\stackrel{-1^3}{\to }0.$$
Let $D(n)$ denote the number of steps of the procedure. Thus $D(100)=4$.

Let $S(N)$ denote the sum of $D(n)$ for all positive integers $n$ strictly less than $N$.

For example, $S(100)=512$.

Find $S({10}^{17})$.

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扣除立方数

从任意正整数$n$开始，每一步依次扣除不超过$n$的最大立方数，直到$n$变为$0$。

例如，从$n=100$开始需要$4$步：
$$100\stackrel{-4^3}{\to }36\stackrel{-3^3}{\to }9\stackrel{-2^3}{\to }1\stackrel{-1^3}{\to }0$$
记$D(n)$为从$n$开始所需要的步数，因此$D(100)=4$。

记$S(N)$为所有严格小于$N$的正整数$n$所对应的$D(n)$之和。

例如，$S(100)=512$。

求$S({10}^{17})$。

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