On a variant of Giuga numbers
  In this paper, we characterize the odd positive integers $n$ satisfying the congruence $\sum_{j=1} ^ {n-1} j^{\frac{n-1}{2}}\equiv 0 \pmod n$. We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8. 
Jose Maria Grau, Florian Luca, Antonio M. Oller-Marcen
math.NT