Limit points and discrepancy of the fractional parts of $b^n/n$

For $b\in \mathbb N$, $b\ge 2$ we determine the limit points of a certain subsequence of $$ \bigg\{\frac{b^n}{n}\ (\bmod{1}):n\in \mathbb N\bigg\}. $$ Further, we obtain an estimate of the discrepancy of the said subsequence, improving on a result of Cilleruelo, Kumchev, Luca, Rué and Shparlinski.