Highly Divisible Number
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:
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 \(20=2\times2\times5\).