Eccoci.

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:
$$ \begin{align} \begin{split} D(1) &= 1 \\ D(p) &= 1, \text{ for any prime } p \\ D(pq) &= D(p)q + pD(q), \text{ for any positive integers } p,q \gt 1. \end{split} \end{align} $$

Now let $\{a_1, a_2,\ldots,a_k\}$ 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 $S(5\times 10^4) \mod 999676999$.