Counting Ordered Factorisations

 Published on Sunday, 13th December 2020, 07:00 am; Solved by 187;
Difficulty rating: 35%

Problem 738

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

\[ n = x_1\times x_2\times x_3\times \ldots\times x_k\qquad 1\le x_1\le x_2\le\ldots\le x_k \]

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