Primes of the form $\ \varphi(n)^{\varphi(n)}+n\ $ or $\ n^n+\varphi(n)\ $ for composite $n$

Question

This question :

Do further prime numbers of the form $n^n+\varphi(n)$ exist?

deals about prime numbers of the form $$n^n+\varphi(n)$$

I know no composite number $n$, such that this expression is prime.

Strangely, I neither know a composite $n$ such that $$\varphi(n)^{\varphi(n)}+n$$ is prime. This expression is prime for $n=1,2,3,7,463$ and no other $n\le 1\ 000$.

Is it a coincidence that I did not find a composite number $n$ such that either expression is a prime, or can it be proven that we never get a prime for composite $n$ ?

Answer 1

Partial answer.

Theorem: Any group of order $n$ is cyclic $\Longleftrightarrow \gcd(n,\varphi(n))=1$

It follows that if there is a non-cyclic group of order $n$, then neither of $n^n+\varphi(n)$ and $\varphi(n)^{\varphi(n)}+n$ is a prime.

This means that $n$ needs to be a product of distinct primes none of which divides another lowered by $1$ (e.g. $15=3\times 5$ with $3\not\mid (5-1)$). See OEIS A003277. Funny fact, although I don't see how that can really be relevant here, that for such numbers, $\varphi(n)^{\varphi(n)}\equiv 1\bmod n$.