For every positive integer $\ n\ $ define $$f(n):=\sum_{j=1}^n j^{j+1}=1+2^3+3^4+4^5+\cdots +n^{n+1}$$
Does $\ n\mid f(n)\ $ hold for infinite many integers $\ n\ $ ?
With PARI/GP , I found the following solutions so far :
? n=1;s=1;while(n<10^6,n=n+1;s=s+n^(n+1);if(Mod(s,n)==0,print1(n," ")))
3 13 16 17 89 208 577 12029 42768 59785 74185 113267
The program is still in progress. The largest prime solution I found so far is $\ 577\ $. Is it the largest ?