We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,\tag{1}$$ with the definition $\operatorname{rad}(1)=1$. You can see this definition and the corresponding properties of this arithmetic function for example from this Wikipedia.
I'm trying to create irrational and transcendental numbers using infinite series and this arithmetic function $(1)$.
Question 1. I've written a proof for the first one, that is a proof of the statement that $\sum_{n=1}^\infty\frac{\operatorname{rad}(n)}{2^{2^n}}$ is a transcendental number. Is my Proof below right? Many thanks.
Claim. The real number $$\sum_{n=1}^\infty\frac{\operatorname{rad}(n)}{2^{2^n}}$$ is a transcendental number.
Proof. We will need in our proof to use the (obvious) inequality $\operatorname{rad}(n)\leq n$, and the following inequalitites that can be proven by mathematical induction: $\forall n\geq 1$ one has $n\leq 2^n$, and also $\forall n\geq 1$ the inequality $2^n-n\geq 2^{n-1}$ holds.
Let $x=\sum_{n=1}^\infty \operatorname{rad}(n)2^{-2^n}$ and we denote $x_m=\sum_{n=1}^m \operatorname{rad}(n)2^{-2^n}$. Then $x_m$ is rational with denominator $2^{2^m}$. Since for any positive integer $k$, $$|x - x_m| \le 2 \cdot \frac{2^m}{2^{2^{m+1}}} < \left(2^{2^{m-1}}\right)^{-k}$$ for sufficiently large $m$, $x$ is a Liouville number and therefore is transcendental by Liouville's theorem.$\square$
On the other hand I would like to know a different and simple example of an irrational number represented as a series involving this arithmetic function (I know that if my proof is right trancendental number$\implies$ irrational number). I mean series of the kind $$\sum_{n=1}^\infty\frac{\operatorname{rad}(n)^\alpha}{s_n},\tag{2}$$ and here $0<\alpha$ is a fixed rational/integer number and $s_n\geq 1$ is the sequence of positive integers that you need to create yourself example.
Question 2. Can you provide me an example of the type $(2)$ showing that it is an irrational number? If you know such example from the literature please refer it, answering this question as a reference request and I search and read such statement from the literature. Many thanks.