Sum of Products

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 $S(5\times 10^4) \mod 999676999$.

First, let’s understand the problem.

It is given that $D(1) = 1$, $D(p) = 1$ for any prime $p$, and $D(pq) = D(p)q + pD(q)$ for any positive integers $p,q > 1$.

The value of a partition $\{a_1, a_2,\ldots,a_k\}$ of a number $n$ is defined as $P=\prod_{j=1}^{k}D(a_j)$.

It is also given that $G(n)$ is the sum of all such $P$ for all possible partitions of $n$.

Finally, it is defined that $S(N)=\sum_{n=1}^{N}G(n)$ and we have to find $S(5\times 10^4) \mod 999676999$.

Let’s break it down.

Clearly D(p) is different for composite and prime numbers. For any prime p, D(p) = 1. For a composite number D(pq), we have D(pq) = D(p)q + pD(q) (where p, q are factors of pq).

This problem can be solved using Dynamic Programming along with number theory concepts.

We start from D(1) to D(n) in a bottom-up manner.

– Initial values of D(i) for i<=5 have been given i.e., D(1)=1, D(2)=1, D(3)=1, D(4)=5, and D(5)=1.

– We also use the Sieve of Eratosthenes to generate primes up to n. For every number, we initialize D(i) as 1 (assuming i is a prime).

– After this, for each number, we update D(i) using the formula: for a composite number D(i) = D(p)q + pD(q) (where p, q are factors of i).

Now comes the calculation of $G(n)$ and $S(N)$.

– G(n) is the sum of $P$ for all partitions of $n$. Hence for each number ‘n’, we have the equation G(n) = G(n-1) + D(n)*G(n-1) (depending on whether n itself can be included once or not). We use the concept of partitions in number theory here – where the partition function $p(n)$ gives the number of distinct ways n can be represented as a sum of natural numbers.

– After we have calculated G(n) for all numbers up to 5*10^4, we calculate the cumulative sum to get S(N).

The calculations are big, hence we need to use modular arithmetic at every step due to integer overflow.

The modulo operator rules ensure that the remainder of a sum or product is the remainder of the sum or product of the remainders, which is used here.

The answer will be $S(5\times 10^4) \mod 999676999$.

Please remember that this is a complex programming problem and requires good understanding of Dynamic Programming (DP) and number theory. Implementing it requires proficient programming skills.

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