HMMT November 2020 U36

Let $p_{i}$ be the $i$ th prime. Let

$$f(x)=\sum_{i=1}^{50} p_{i} x^{i-1}=2+3 x+\cdots+229 x^{49}$$

If $a$ is the unique positive real number with $f(a)=100$, estimate $A=\lfloor 100000 a\rfloor$. An estimate of $E$ will earn $\max (0,\lfloor 20-|A-E| / 250\rfloor)$ points.