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Problem 688

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Piles of Plates

We stack $n$ plates into $k$ non-empty piles where each pile is a different size. Define $f(n,k)$ to be the maximum number of plates possible in the smallest pile. For example when $n = 10$ and $k = 3$ the piles $2,3,5$ is the best that can be done and so $f(10,3) = 2$. It is impossible to divide $10$ into $5$ non-empty differently-sized piles and hence $f(10,5) = 0$.

Define $F(n)$ to be the sum of $f(n,k)$ for all possible pile sizes $k\ge 1$. For example $F(100) = 275$.

Further define $S(N) = \displaystyle\sum_{n=1}^N F(n)$. You are given $S(100) = 12656$.

Find $S(10^{16})$ giving your answer modulo $1\ 000\ 000\ 007$.

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盘子分堆

将$n$个盘子分为$k$堆，每一堆既不能为空，也不能与另一堆有相同数量的盘子，并记$f(n,k)$为所有分法中最少的那堆可能分得的最大盘子数量。例如，若$n = 10$，$k = 3$，那么最理想的分法是$2,3,5$，因此$f(10,3) = 2$。由于无法将$10$个盘子按照上述要求分为$5$堆，因此$f(10,5) = 0$。

记$F(n)$为所有堆数$k\ge 1$的$f(n,k)$之和。例如，$F(100) = 275$。

再记$S(N) = \displaystyle\sum_{n=1}^N F(n)$。已知$S(100) = 12656$。

求$S(10^{16})$并将你的答案对$1\ 000\ 000\ 007$取余。

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