Sequences for that $\sum_{n} \frac{1}{x_n}$ is divergent and $\sum_{n} \frac{1}{x_n \ln x_n}$ is convergent

We will denote with $(x_n)$ a given sequence and we introduce the following two series.

$$S^* = \sum_{n} \frac{1}{x_n} \quad \text{and} \quad S_* = \sum_{n} \frac{1}{x_n \ln x_n}.$$

We know that if $(x_n)$ are for example the Fibonacci numbers greater then $1$, then $S^*$ and $S_*$ are convergent. If $(x_n)$ are the prime numbers then $S^*$ is divergent and $S_*$ is convergent. If $(x_n)$ are the natural numbers greater then $2$, then both series are divergent.

Question. How could we characterise the $(x_n)$ sequences, for that $S^*$ is divergent and $S_*$ is convergent. I would be also glad to see some reference in this topic. If we cannot characterise $(x_n)$ then is there any special property of such sequences?

I don't think you're going to find any characterization that's simpler than your definition itself. The boundary between convergence and divergence is very delicate. For example, if $x_n = n( \log n )(\log\log n) (\log\log\log n)^\alpha$, then $S_*$ converges regardless of the value of $\alpha$, but $S^*$ converges when $\alpha>1$ and diverges when $\alpha\le1$.