# The Game of Numbers (#02)

### Related to The Game of Numbers (#01)

Agatha just decided it still isn't enough because she thinks it's way too easy. She decides to add one more rule to the list and change some of the existing rules.

• Pick a number between $$7$$ and $$100$$, inclusive.
• Take the prime factorization of that number, in exponential form (i.e. express it as $$p_1^{e_1}p_2^{e_2}...$$). Take all the $$p_i$$ and $$e_i$$, and choose either their sum or their product: your number becomes the chosen value.
• Keep doing this until either your number becomes less than $$7$$, you end up with a number that you have already ended up with in the game, or go over your limit.
• You lose if your opponent has a longer turn than you in the same round, and your limit (on starting numbers) is now increased by $$20$$ instead of $$10$$. In case of a tie, simply repeat the round.
• NEW RULE: Any number you end up with as a result of this process CANNOT be picked anymore, to avoid spamming numbers like $$72$$ and $$155$$.

She also decided to just make it longer to give numbers like 210 a chance, so you have 15 rounds instead of 10 to work with.
QUESTION: What are the new ideal numbers?

BONUS QUESTION: Assuming that you understand this, if Agatha goes first, what are the best counters against her first numbers?