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Theorema 19 in Euler's memoir "Variae observationes circa series infinitas" says

The sum of the reciprocals of the prime numbers is infinitely great but is infinitely times less than the sum of the harmonic series. And the sum of the former is as the logarithm of the sum of the latter.

In modern terms ($p$ denotes a prime number) $$\lim_{n\to \infty}\left(\sum_{k\le n}\frac{1}{k}-\ln{n}\right )=\gamma_1,$$ and $$\lim_{n\to \infty}\left(\sum_{p\le n}\frac{1}{p}-\ln{\ln{{n}}}\right )=\gamma_2,$$ where $\gamma_1\approx 0.5772156649$ is the Euler–Mascheroni constant and $\gamma_2\approx 0.2614972128$ is the Mertens Constant.

Can we extend this further? That is can we indicate a subseries of the harmonic series whose sum is "infinitely times less" than the sum of the Euler series and "equals" the logarithm of the sum of the latter: $$\lim_{n\to \infty}\left(\sum_{k(with\,some\,condition\,on\,it)\le n}\frac{1}{k}-\ln{\ln{{\ln{n}}}}\right )=\gamma_3?$$

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    $\begingroup$ Integers of the form $\lfloor n \log{n} \log\log{n}\rfloor$ should do. $\endgroup$
    – alpoge
    Feb 25, 2015 at 10:12
  • $\begingroup$ Maybe primes of order of primality, i.e such that the smallest $m$ such that the $m$-the iterate of the prime counting function, is at least equal to $2$. $\endgroup$ Feb 25, 2015 at 15:56
  • $\begingroup$ Interestingly, the sum of reciprocals of the super-primes (prime-index primes) converges to $\sim 1.04$, see pp.10-12 in the article A43 in integers-ejcnt.org/vol13.html (New Bounds and Computations on Prime-Indexed Primes, by J. Bayless, D. Klyve, and T.O. e Silva). $\endgroup$ Feb 26, 2015 at 4:44
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    $\begingroup$ How about primes with a prime number of digits? $\endgroup$
    – Lucia
    Mar 9, 2015 at 14:10

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