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Piles of Plates

Published on Sunday, 10th November 2019, 01:00 am; Solved by 287

Problem 688

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 $1\,000\,000\,007$.