# Highly Divisible Number

There is a positive integer $$N$$ with $$10000$$ (not necessarily distinct) prime factors that allows for the following recursive process to occur:

Let us define $$P(n$$) as the function that counts the number of prime factors that $$n$$ has. For example, $$P(N)=10000$$. Let us also define the following recursive relation:

$\large a_{n+1}=\frac{a_n}{P(a_n)}$

If we let $$a_0=N$$, $$a_n$$ is always a positive integer, and for some positive integer $$k$$, $$a_n=2$$ for all $$n\geq k$$. What is the smallest prime number not found in the prime factorization of $$N$$?

Note: When I say the number of prime factors, I mean the total number of prime numbers needed to make up the number itself. For example, $$P(20)=3$$ because $$20=2\times2\times5$$.

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