What is the Dirichlet serie of The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$?

Question

The function $A$ (OEIS A001414) which gives the sum of prime factors (with repetition) of a number $n$ and defined by $$ A(n)=\sum \limits_{p^{\alpha}\parallel n}\alpha p $$ is this serie calculated before ? and where i can found the resources $$ \sum \limits_{n\geq 1} \frac{A(n)}{n^s} $$ and $$ \sum \limits_{n\geq 1} \frac{\mu(n)A(n)}{n^s} $$

Answer 1

Unclear what you are asking for exactly. With $P(s)=\sum_{p^k}p^{1-sk}=\sum_p \frac{p^{1-s}}{1-p^{-s}}$ $$\sum_{n\ge 1} n^{-s}\sum_{p^k| n} p = \zeta(s)P(s)$$

$$\sum_{n\ge 1} n^{-s}\mu(n)\sum_{p^k| n} p = \sum_{p^k} p \sum_{n\ge 1} (p^k n)^{-s}\mu(p^k n)$$ $$=-\sum_p p \sum_{n\ge 1,p\,\nmid \,n} (p n)^{-s}\mu(n) = -\sum_p p \frac{p^{-s}}{(1-p^{-s})\zeta(s)}= -\frac{P(s)}{\zeta(s)}$$