A partition of $n$ is a set of positive integers for which the sum equals $n$.
The partitions of 5 are:
$\{5\},\{1,4\},\{2,3\},\{1,1,3\},\{1,2,2\},\{1,1,1,2\}$ and $\{1,1,1,1,1\}$.

Further we define the function $D(p)$ as:
$$\begin{array}{r}\begin{array}{rl}D(1)& =1\\ D(p)& =1,\text{ for any prime }p\\ D(pq)& =D(p)q+pD(q),\text{ for any positive integers }p,q>1.\end{array}\end{array}$$

Now let $\{a_1,a_2,\dots ,a_k\}$ be a partition of $n$.
We assign to this particular partition the value:
$$P=\prod _{j=1}^{k}D(a_j).$$

$G(n)$ is the sum of $P$ for all partitions of $n$.
We can verify that $G(10)=164$.

We also define: $$S(N)=\sum _{n=1}^{N}G(n).$$ You are given $S(10)=396$.
Find $S(5\times {10}^4)\mathrm{mod}999676999$.