It is well-known in high-level mathematics that:

$\displaystyle \lim_{n\to\infty}\left[\left(\sum_{k=1}^n\frac{1}{k}\right)-\ln(n)\right]=\gamma$ (Euler-Mascheroni Constant)

$\displaystyle \lim_{n\to\infty}\left[\left(\sum_{p~prime}^n\frac{1}{p}\right)-\ln(\ln(n))\right]=M$ (Meissel-Mertens Constant)

Is there a divergent series $\displaystyle \sum a_k$ such that:

$\displaystyle \lim_{n\to\infty}\left[\left(\sum_{k=1}^n a_k\right)-\ln(\ln(\ln(n)))\right]=constant$

If so, can it be generalized to $\displaystyle \ln(\ln(...(\ln(n))))$?