Vtyjkyuk

Can the expression $$n^n+varphi(n)$$ be a prime number for some integer $n>19$ ?

For $n=1,2,3,19$ and no other positive integer $nle 3 000$, the expression is prime. A further prime of the desired form must have more than $10 000$ digits. For $n>2$, only odd $n$ need to be considered because $varphi(n)$ is even for $n>2$

Moreover, I search a prime factor of the composite $283$-digit number $$f(133)=133^133+108$$ Probably , there is no prime factor with $20$ digits or less.

Since you found no prime with composite $n$ for $n leq 10^4$ the only explanation that I am aware of is that $phi(n)$ is very likely to have form $phi(n)=2r(n)$, where $r(n)$ is either a prime dividing $n$ or is divisible by a prime that also divides $n$.

So, further study of function $phi$ is needed, that is, to determine for which $n$ we have that there is a prime $p$ that divides both $n$ and $phi(n)$.

It could be that quite often we have $phi (n)=m(n) cdot p(n)$, where $m(n)$ is even and $p(n)$ is some prime dividing $n$, because then we could not have $phi(n)=m(n) cdot p(n) | (n-1)$ because of $p(n)|n$ and that would be in favour of Lehmer´s conjecture.