On a certain arithmetic function defined via Bernoulli numbers

We investigate the integrality property of an arithmetic function defined on the set of odd numbers $n\geq 3$. It is constructed via Bernoulli numbers. As a result, we show that this function unifies three distinct number classes -- primes, Carmichael numbers, and Giuga numbers -- into a single integrality criterion. The article is elementary and therefore accessible to a broad audience.