An Enormous Factorial

For any prime $p$ the number $N(p, q)$ is defined by
$N(p, q) = \sum_{n = 0}^q T_n \cdot p^n$
with $T_n$ generated by the following random number generator:

$S_0 = 290797$
$S_{n + 1} = S_n^2 \bmod 50515093$
$T_n = S_n \bmod p$

Let $\operatorname{Nfac}(p, q)$ be the factorial of $N(p, q)$.
Let $\operatorname{NF}(p, q)$ be the number of factors $p$ in $\operatorname{Nfac}(p, q)$.

You are given that $\operatorname{NF}(3,10000) \bmod 3^{20} = 624955285$.

Find $\operatorname{NF}(61, 10^7) \bmod 61^{10}$.

This seems like a very specialized problem that probably comes from a specific context, such as a programming contest or a number theory problem set, which generally requires a higher mathematical knowledge including understanding of Number Theory and Modular Arithmetic. Since it involves generation of pseudorandom numbers and advanced mathematical theories, it’s a bit complicated.

First, we need to understand the definitions. $N(p, q)$ is a sum with each term being $p^n$ times the residue of a pseudorandom term modulo $p$. This pseudorandom term is generated by a recursive equation modulo a large number. $Nfac(p, q)$ is defined as the factorial of $N(p, q)$, and $NF(p, q)$ counts the number of prime factors $p$ in $Nfac(p, q)$.

Considering the nature of this problem, here’s a general approach you could take:

1. First, write a function that emulates the given pseudorandom generator that produces $T_n$ values.

2. Write another function that calculates $N(p, q)$ using these pseudorandom values.

3. Use the above function to calculate $Nfac(p, q)$.

4. Finally, calculate $NF(p, q)$ by counting the number of prime factors $p$ in the result of the previous step.

This problem cannot be solved analytically – it requires programming in order to generate the sequence of $T_n$ and $N(p, q)$. You also have to use a clever way to count the factors, as the straightforward approach is going to be too slow for large numbers.

However, this is beyond the capabilities of this AI and would be best tackled using a programming language that can handle large numbers and have efficient algorithms for factoring numbers, like Python or C++.

Keep in mind that this problem inherently has difficulty as it combines concepts from number theory and pseudorandom sequences. It may be a problem aimed toward a more advanced audience with a good programming background.

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