# Parity Coefficient

If $$\dbinom{n}{r_n}$$ is an even number for natural numbers $$r_n < n$$ and denote $$R_n$$ as the smallest possible value of $$r_n$$, find the value of $$n$$ less than $$10^4$$ such that its corresponding $$R_n$$ is maximized.

Details and Assumptions:

• As an explicit example, $${13 \choose 2}, {13 \choose 3}, {13 \choose 6}, {13 \choose 7}, {13 \choose 10}, {13 \choose 11}$$ are all even numbers. So the possible values of $$r_{13}$$ are $$2,3,6,7,10,11$$. With $$R_{13} = 2$$.
• Note that there doesn't exist some values of $$r_n$$. For example, $${7 \choose 1}, { 7 \choose 2} , \ldots , {7 \choose 6}$$ are all odd numbers. So there's no value of $$r_7$$ nor $$R_7$$.

Bonus: Can you generalize this for some constant $$M$$ such that $$n < M$$?

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