On a variant of Giuga numbers

Abstract

In this paper, we characterize the odd positive integers $n$ satisfying the congruence ${\sum }_{j=1}^{n-1}{j}^{\frac{n-1}{2}}\equiv 0(\mathrm{mod}n)$. We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8.

14 pages