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Problem 738

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Counting Ordered Factorisations

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 $1\ 000\ 000\ 007$.

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有序因数分解计数

记$d(n,k)$为将$n$表示为$k$个递增整数之积的方式数
$$n = x_1\times x_2\times x_3\times \ldots\times x_k\qquad 1\le x_1\le x_2\le\ldots\le x_k$$

再记$D(N,K)$为所有$1\le n\le N$和$1\le k\le K$对应的$d(n,k)$之和。

已知$D(10, 10) = 153$，$D(100, 100) = 35384$。

求$D(10^{10},10^{10})$，并将你的答案对$1\ 000\ 000\ 007$取余。

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