Formulation in terms of Bernoulli numbers
where is the Bernoulli number corresponding to .
Formulation in terms of power sums
This formulation dates to Giuga in the 1950s.
Formulation in terms of Giuga numbers
There is no (composite) natural number that is both a Carmichael number and a Giuga number. Note that this is equivalent to the power sums formulation because composite number satisfies the power sums condition iff it is both a Carmichael number and a Giuga number.