What are the odds of it?

Let $p$ be the probability of winning a given Rock Paper Scissors game. Then $\begin{aligned} p &= \overbrace{\frac{1}{3}}^{\color{#3D99F6}\text{Probability you win immediately}} + \overbrace{\frac{1}{3}p}^{\color{#3D99F6}\text{Probability you win after drawing}}\\ \frac{2}{3} p &= \frac{1}{3} \\ p &= \frac{1}{2} \end{aligned}$

Now, we solve $\begin{aligned} p^n &= \frac{1}{145289} \\ \frac{1}{2^n} &= \frac{1}{145289} \\ 2^n &= 145289 \\ n &\approx 17.15 \end{aligned}$

Thus $\lfloor n\rfloor = \boxed{17}$ .