Examples of integer sequences that have a distribution approx $1/\log(n)$, like the primes do?

Question

It is well known that the primes are distributed such that they occur with an approximate "likelihood" of $1/\log(n)$ around the integer $n$ - or more precisely, the number of primes up to $n$ is $$\pi(n) \sim \int_2^n \frac{dt}{\log(t)}.$$

Question: Are there other sequences that have a distribution such that the likelihood of $n$ being a member of that sequence is approx $1/\log(n)$?

Further Question: What properties or constraints would such sequences need to adhere to?

Answer 1

One such sequence is $a_n = \lfloor n\log n\rfloor$. (In a similar way, one can construct sequences with any desired density.)

Another is $\{n\in\Bbb N\colon n$ is divisible by $\lfloor \log n\rfloor\}$ (which also has obvious generalizations).

Answer 2

There is a lot of such sequences. For example:

$p_n +1$

$p_n +7$

$p_n + (-1)^n$

Answer 3

You can construct a lot of sequences from the primes themselves that have this density. For instance, if you exclude all primes of the form $n^2+1$, since the density of the perfect squares decreases faster than that of the primes, you still asymptotically have density $\frac1{\log n}$. (In fact, in this case it’s not known whether infinitely many primes are excluded; see Primes of the form $n^2+1$ - hard?)

Or, perhaps slightly more interesting: The density of $k$-almost primes (i.e. integers with $k$ prime factors) is asymptotic to

$$ \frac{(\log\log n)^{k-1}}{(k-1)!\log n} $$

(see asymptotic density of $k$-almost primes). You can combine this with Greg Martin’s approach: The sequence you get from the sequence of $k$-almost primes by omitting $\left\lfloor\frac{(\log\log n)^{k-1}}{(k-1)!}\right\rfloor$ terms after a term $n$ has density $\frac1{\log n}$.