It is known and quite easy to prove that $S_{\mathbb N}(x) = \sum_{n\in\mathbb N, n\leq x} \frac 1 n$ grows as $\ln x$. Even more, $\lim_{x\rightarrow\infty} S_{\mathbb N}(x)-\ln x = \gamma$, the Euler-Mascheroni constant.

It was also proved by Mertens that $S_{\mathbb P}(x)$, the sum of reciprocals of the primes not exceeding $x$, grows as $\ln\ln x$, and actually, $\lim_{x\rightarrow\infty} S_{\mathbb P}(x)-\ln\ln x = M$, the Meissel-Mertens constant.

My questions is, is there an example of a set $A\subset \mathbb N$, such that $\lim_{x\rightarrow\infty} S_A(x)-\ln\ln\ln x=C$ for some real $C$. It is quite obvious that such a set exists, however I am looking for some "natural" examples (maybe some that were not specifically constructed as an answer to this question, but rather appeared during some research).

on averagegrow like $2p_n\log\log p_n$, but its individual values fly around and are often negative. $\endgroup$ – Yaakov Baruch Oct 31 '19 at 13:35