Define $d(n,k)$ to be the number of ways to write $n$ as a product of $k$ ordered integers

$$n=x_1\times x_2\times x_3\times \dots \times x_{k}1\le x_1\le x_2\le \dots \le x_k$$

Further define $D(N,K)$ to be the sum of $d(n,k)$ for $1\le n\le N$ and $1\le k\le K$.

You are given that $D(10,10)=153$ and $D(100,100)=35384$.

Find $D({10}^{10},{10}^{10})$ giving your answer modulo $1000000007$.