Define $$f(n):=\sum_{j=1}^n j^{n+1-j}=1+2^{n-1}+3^{n-2}+\cdots+(n-1)^2+n$$ for a positive integer $n$.
With PARI/GP, this function can be calculated with the self-defined function
f(n)=sum(j=1,n,j^(n+1-j))
Question $1$ : Is $f(n)$ only a perfect power for $n=3$ ?
I checked upto $n=10^4$ and only found one perfect power, namely for $n=3$. I do not even have an idea for the perfect squares. The PARI/GP-code for those who want to extend the search limit :
gp > for(m=1,10^4,if(ispower(f(m))>0,print1(m," ")))
3
gp >
Question $2$ : If $f(n)$ only prime for $n=2$ and $n=34$ ?
Upto $n=7\ 400$ , there are no other primes $f(n)$.
It seems that $f(n)$ cannot be easily calculated with a formula. So, I think I cannot use fast primality testing tools like PFGW, but someone might have a faster software for testing general numbers.
The first such number from which I do not know a prime factor, is $f(138)$. So, small factors are not forced, hence I guess we can only search for further primes.