It is known that there is a unsolved problem related to primes of the form $n^n+1$ as is explained in page 160 of [1] (see also page 156, and the OEIS page related to this integer sequence A121270). In this post I ask about if we can to get a characterization for this sequence of primes in terms of (some) number-theoretic functions.
Question. I would like to get a characterization of these primes $n^n+1$ in terms of number-theoretic functions (see my attempts) It is required that your characterization is a $\iff$ statement.
My attempt was the following deductions (while which I require is a $\iff$ statement: a characterization in terms of particular values of common arithmetic functions in number theory).
Claim 1. If $n^n+1$ is a prime number, then $$\varphi(\varphi(n^n+1))=\frac{\varphi(n)}{n}(\psi(n^n+1)-2)\tag{1}$$ holds, where $\varphi(k)$ denotes the Euler's Totient Function and $\psi(k)$ denotes the Dedekind Psi Function.
Claim 2. If $n^n+1$ is prime, then the equations
$$\varphi(\varphi(A))^B=(A-1)^{B-1}\cdot\varphi(B)^B$$
and $$\psi(A-1)^B=\psi(B)^B\cdot(A-1)^{B-1}$$
holds for some choice of integers $A,B\geq 1$ (take $B=n=\sqrt[B]{A-1}$).
Computation evidence for Claim 2. I've tested with a GP program (Sage Cell Server) that the only integers that satisfy both equations are $A=2,5$ and $257$ when the variables run over the integers $2\leq A\leq 10^3$ and $1\leq B\leq 10^3$
References:
[1] Michael Křižek, Florian Luca and Lawrence Somer, 17 Lectures on Fermat Numbers, CMS Books in Mathematics, Canadian Mathematical Society, Springer-Verlag (2001).