# An integer sequence defined from a variation of the Lucas–Lehmer primality test: the case of the Euler's totient function

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.

• If some user is interested maybe the case involving the sum of divisor function can be studied, – user243301 Mar 15 '18 at 17:09
• Many thanks @MichaelHardy – user243301 Mar 15 '18 at 17:54
• well... Sometimes I do substantial MathJax edits, but in this case I seem to have done the most trivial possible edit while deciding to leave the rest as it was, so maybe you exaggerate. – Michael Hardy Mar 15 '18 at 17:57
• Okey... I've exaggerated $\checkmark$ @MichaelHardy – user243301 Mar 15 '18 at 18:53
• Wow, it sure isn't in the OEIS. I tried searching for just 4,6,10,38 and even that got no results. – Mr. Brooks Mar 15 '18 at 23:08