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Based on Erdos idea, a subset of natural numbers is "dense" if the sum of its reciprocals is infinite, in such case it contains an infinite set of arithmetic progressions.

Prime numbers are "dense" in that sense, sum of reciprocals is unbound but grows slowly, as log log(n). List of twin primes is not dense, as the limit is Brun's constant.

I am interested in what is in the middle: what additional conditions can we apply to the prime numbers, that a sum is still unbound, but it grows much much slower, say, as log log log log(n), or, on the contrary, became bound, but the limit is huge so a sequence is "almost" dense.

Obviously, some subsets are too sparse (Mersenne primes, for example). Some criteria are not interesting if they involve digits base 10 - this is just a numerology (for example, delicate primes). Some good candidates are lucky primes, good primes, balanced prims, and may be the others.

What is known about the sum of the reciprocals of these sequences?

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  • $\begingroup$ Reads like a fishing expedition. $\endgroup$
    – Gerry Myerson
    Commented Dec 27, 2021 at 14:50
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    $\begingroup$ How about the set of all primes $p$ such that there is at least one prime other than $p$ between $p-100$ and $p+100$? I'm guessing the convergence argument for twin primes could be strengthened to apply to such primes, giving a finite-but-large sum. $\endgroup$
    – Gerry Myerson
    Commented Dec 27, 2021 at 14:56
  • $\begingroup$ Thank you, I was also thinking about almost the same. Taking n=100 we get some limit B(100). As n increases, B(n) increases as well. Full set of primes is a limit B(inf). however, it would be much more interesting to have a function F with a property that F(n<666) (or some other value) is finite, while F(667) is not. N(n) is always finite for finite n, so we cant "tackle" the border. $\endgroup$
    – tzimie
    Commented Dec 27, 2021 at 15:10
  • $\begingroup$ $\sum \frac{1}{p\log\log p}$. $\endgroup$
    – mathworker21
    Commented Dec 27, 2021 at 16:10

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