For any integer $m>2$, let $P_m$ be the set of primes less than $m$, and let $$ f(m) = \sum\limits_{p \in P_m} \frac{1}{m-p}. $$ For example, $f(3)=\frac{1}{3-2}=1$, $f(4)=\frac{1}{4-2}+\frac{1}{4-3}=\frac{3}{2}$, and so on.

The question is to estimate $I=\inf\limits_{m>2} f(m)$.

A simple Mathematica calculation shows that $f(m)\geq f(223)\approx 0.60178$ for all $m$ up to $10,000$. It is true that $I>0$? Is $I>0.5$? Is $I=f(223)$?