956. Super Duper Sum

The total number of prime factors of $n$$n$, counted with multiplicity, is denoted $\mathrm{\Omega }(n)$$\Omega(n)$.
For example, $\mathrm{\Omega }(12)=3$$\Omega(12)=3$, counting the factor $2$$2$ twice, and the factor $3$$3$ once.

Define $D(n,m)$$D(n, m)$ to be the sum of all divisors $d$$d$ of $n$$n$ where $\mathrm{\Omega }(d)$$\Omega(d)$ is divisible by $m$$m$.
For example, $D(24,3)=1+8+12=21$$D(24, 3)=1+8+12=21$.

The superfactorial of $n$$n$, often written as $n\mathrm{\$}$$n\$$, is defined as the product of the first $n$$n$ factorials:

The superduperfactorial of $n$$n$, we write as $n★$$n\bigstar$, is defined as the product of the first $n$$n$ superfactorials:

You are given $D(6★,6)=6368195719791280$$D(6\bigstar, 6)=6368195719791280$.

Find $D(1000★,1000)$$D(1\,000\bigstar, 1\,000)$. Give your answer modulo $999999001$$999\,999\,001$.

956. 超酷阶乘的求和

记 $\mathrm{\Omega }(n)$$\Omega(n)$ 为 $n$$n$ 的质因子的数量（重复质因数计多次）。例如，因为 $12=2^2\times 3$$12 = 2^2 \times 3$，所以 $\mathrm{\Omega }(12)=3$$\Omega(12) = 3$。

记 $D(n,m)$$D(n, m)$ 为：所有满足 $d$$d$ 能整除 $n$$n$、$m$$m$ 能整除 $\mathrm{\Omega }(d)$$\Omega(d)$ 的整数 $d$$d$ 的和。例如：$D(24,3)=1+8+12=21$$D(24, 3)=1+8+12=21$。

我们将前 $n$$n$ 个阶乘之积称作 $n$$n$ 的 超阶乘，记作 $n\mathrm{\$}$$n \$$：

我们将前 $n$$n$ 个超阶乘之积称作 $n$$n$ 的 超酷阶乘，记作 $n★$$n \bigstar$：

已知 $D(6★,6)=6368195719791280$$D(6\bigstar, 6)=6368195719791280$。

求 $D(1000★,1000)$$D(1\,000\bigstar, 1\,000)$ 模 $999999001$$999\,999\,001$ 的值。

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