For which integers $n \in\{1,2, \ldots, 15\}$ is $n^{n}+1$ a prime number?
$n=1$ works. If $n$ has an odd prime factor, you can factor, and this is simulated also by $n=8$ : \[ a^{2 k+1}+1=(a+1)\left(\sum_{i=0}^{2 k}(-a)^{i}\right) \] with both parts larger than one when $a>1$ and $k>0$. So it remains to check 2 and 4 , which work. Thus the answers are $1,2,4$. $\fbox{1,2,4}$.