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

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