I did a variation of the so-called Lucas–Lehmer primality test, I say this Wikipedia. I've used the Euler's totient function $$\varphi(n)=n\prod_{\substack{p\mid n\\ p\text{ prime}}}\left(1-\frac{1}{p} \right)$$ for $n> 1$, and taking $\varphi(1)=1$ as definition. This is a well-known an important multiplicative function.
Our definition is $$\left. \begin{array}{l} E_i=\varphi(E_{i-1})E_{i-1}-2,\quad\text{for }i\geq 1\\ E_0=4 \end{array} \right\}\tag{1}$$
From it, our sequence starts as $$4, 6, 10, 38, 682, 204598,20929966202\ldots\tag{2}$$
Question. Please prove, provide heuristic, or refute the following conjecture:
Conjecture-E: One has that $E_k$ is a square-free integer $\forall k\geq 1$.
Many thanks.
That is, each term of our sequence $E_k$ has no repeated prime factors when $k\geq 1$. It seems that this sequences isn't in the OEIS.