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Let $G_r =$ http://oeis.org/A005250, and $P_r =$ http://oeis.org/A002386.

$\sum_{n=1}^{\infty}{\frac{1}{G_r}} = c_1$

$\sum_{n=1}^{\infty}{\frac{1}{P_r}} = c_2$

Do the constants c_1 and c_2 exist?

The reason I ask is how rare these numbers seems to be. There are only 80 of them known $< 2^{64}$ yet $\infty$ many of them. I kinda expect the sum with gaps to diverge while the primes to converge. The sum of the first 80 gaps is 2.55968417154317 while the first 5 terms added to 2.04166666666667. For the primes, they seem to converge in a spreadsheet to 1.04470058508119. Because this not even a 64-bit spreadsheet, I do not trust it to answer my guess.

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    $\begingroup$ The divergence of the first sum would imply that $G_n$ is $o(n^2)$. The OEIS link mentions that even the weaker $O(n^2)$ is only a conjecture. $\endgroup$
    – Yaakov Baruch
    Commented Dec 19, 2021 at 20:46
  • $\begingroup$ @YaakovBaruch Thanks for the comment. I will say that with the spreadsheet data, the growth of $G_n << n^2$. $\endgroup$
    – John Nicholson
    Commented Dec 20, 2021 at 2:28
  • $\begingroup$ Often it is precisely such data that leads to these conjectures. Rigorously proving such claims is considerably more difficult. $\endgroup$
    – Joshua Stucky
    Commented Dec 20, 2021 at 5:49

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