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 $D(2^N, 2^N)$.