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: $$\lim_{n\to \infty}\left(\sum_{k(with\,some\,condition\,on\,it)\le n}\frac{1}{k}-\ln{\ln{{\ln{n}}}}\right )=\gamma_3?$$