TITLE: 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 [5 upvotes]: 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 $\log \log \log n$ ?

REPLY [2 votes]: $$k_n=1+\left\lfloor n\cdot\log n\cdot\log\log n\right\rfloor$$
Proof:
$\dfrac{\mathrm d}{\mathrm dx}\log\log\log x=\dfrac1{x\cdot\log x\cdot\log\log x}$.