Giuga number

Definition

A composite number $n$ is termed a Giuga number if and only if it satisfies the following equivalent conditions:

For every prime number $p$ dividing $n$ , we have that $p$ divides $(n/p)-1$ , or equivalently, that $p^{2}$ divides $n-p$ .
We have the following congruence:

$\sum _{i=1}^{n-1}i^{\varphi (n)}\equiv -1{\pmod {n}}$

where $\varphi (n)$ denotes the Euler totient function of $n$ .

Note that all prime numbers satisfy the stated condition but we deliberately exclude them by imposing the restriction of being composite.