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 $1000000007$.