Is there a "natural" subsequence of positive integers $k_1 < k_2 < \ldots$ such that $\sum_{i=1}^n \frac{1}{k_i} = \Theta (\log \log \log n)$?

Question

The harmonic series partial sums grow like $\log n$, and the sum of inverses of the first $n$ primes grows like $\log \log n$. Is there an example of a "nautral" subset of the positive integers (say a subset defined by some interesting property, like being prime for example) such that the sum of inverses of the first $n$ numbers grows like $\log \log \log n$ ?

Answer 1

$$k_n=1+\left\lfloor n\cdot\log n\cdot\log\log n\right\rfloor$$ Proof: $\dfrac{\mathrm d}{\mathrm dx}\log\log\log x=\dfrac1{x\cdot\log x\cdot\log\log x}$.