On a certain arithmetic function defined via Bernoulli numbers

Abstract

The paper defines a certain function \(χ(n)\) on the set of odd integers \(n \ge 3\). The integrality property of this function is investigated. The main result establishes that the function is integer-valued if and only if \(n\) is a prime number or an odd Giuga number. Thus, the integrality of \(χ(n)\) serves as a single criterion unifying three classes: prime numbers, Carmichael numbers, and odd Giuga numbers. The proof relies exclusively on the von Staudt--Clausen theorem on the denominators of Bernoulli numbers and Korselt's criterion for Carmichael numbers, which makes the exposition elementary. A connection is proven between prime numbers \(p\) for which \(χ(p)=p-1\) and Novák--Carmichael numbers. It is proven that the primes satisfying this criterion form a sequence with the identifier A337119 in the Online Encyclopedia of Integer Sequences. For the classes \(P_μ\) of primes with a fixed value \(χ(p)=μ\), a characterization is obtained in terms of the exponents of the prime divisors of the numerator of the corresponding Bernoulli number \(B_{p-1}\). For some \(μ\), explicit congruences are provided. The work is accessible to a wide readership due to its minimal use of advanced mathematical apparatus.

9 pages