Is there a series that is asymptotic to $\log(\log(\log x))$

Question

We know: $\sum\limits_{n=1}^x\dfrac{1}{n}\sim\log(x)$, $\sum\limits_{n=1}^x\dfrac{1}{\text{prime}(n)}\sim\log(\log(x))$.

I wonder if there is another interesting series that diverges even slower than reciprocals of primes, at a rate of $\log(\log(\log x))$. Or perhaps if there is a family of series that diverge ever more slowly, that are asymptotic to $\log(x)$ nested once, twice, thrice, etc.

Answer 1

By the prime number theorem, $\text{prime(n)}\sim n\log(n)$. Therefore

$$\sum^x_{n=1}\frac{1}{\text{prime}(n)\log\Big(\frac{\text{prime}(n)}{n}\Big)}\sim \int_3^x \frac{1}{n\log(n)\log(\log(n))}\,dn\sim\log(\log(\log(x)))$$ In a similar way, you can obtain more examples (though not so "interesting" I guess...).