In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-mangoldt function. I am looking for an analogous arithmetic function of the logarithm function which is of the form $$\beta(n)=\sum_{p_{i}^{\alpha_{i}}\mid n}\alpha_{i}\cdot \log{\log{p_{i}}}$$ where $n=p_{1}^{\alpha_{1}}\cdots p_{s}^{\alpha_{s}}.$ Is this function well known in literature?

P.S. I am sorry if this post doesn't suit for math overflow and would delete it if suggested.

In number theoretical estimations, often we take the logarithms of a natural number to express it properly. A perfect example of this is the von-Mangoldt function. I am looking for an analogous arithmetic function of the logarithm function which is of the form $$\beta(n)=\sum_{p_{i}^{\alpha_{i}}\mid n}\alpha_{i}\cdot \log{\log{p_{i}}}$$ where $n=p_{1}^{\alpha_{1}}\cdots p_{s}^{\alpha_{s}}.$ Is this function well known in literature?

P.S. I am sorry if this post doesn't suit for MathOverflow and would delete it if suggested.