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).

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    $\begingroup$ Are there are "natural" sets with density $1/\log(X)\log\log(X)$? $\endgroup$ – MyNinthAccount Oct 30 '19 at 19:14
  • $\begingroup$ Maybe: the primes $q$ such that $q-1$ has no odd prime divisor up to $C\log(q)$ (unsure what the constant $C$ should be). $\endgroup$ – MyNinthAccount Oct 30 '19 at 19:24
  • $\begingroup$ Maybe: the $p$th primes where $p$ itself is prime? E.g. 3, 5, 11, 17, 31, ... $\endgroup$ – Aeryk Oct 31 '19 at 2:40
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    $\begingroup$ @Aeryk as it turns out, the $n$th term of that series is asymptotically $p_n \log p_n \sim n \log^2 n$, and so the sum of its reciprocals converges. $\endgroup$ – Greg Martin Oct 31 '19 at 7:33
  • $\begingroup$ This does not work, but perhaps something along the lines might: I think that the integer series $np_{p_n}-p_n^2-(n-1)p_{p_{n-1}}+p_{n-1}^2$ may on average grow 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

A nice example was found by Erdos; "On a problem of G. Golomb". 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).

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