Highly Divisible Number

Computer Science Level 5

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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