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).



A very slowly diverging series
A nice example was found by Erdos; "[On a problem of G. Golomb](https://www.renyi.hu/~p_erdos/1961-10.pdf)". Let $p\_1 = 3$, and for $i > 1$, let $p\_i$ be the least odd prime exceeding $p\_{i-1}$ which is not congruent to $1$ mod $p\_{j}$ for any $j < i$. That this sequence of primes has the property you seek follows from eq. (37) there (take logarithms).


