Résumé
Let n be a positive integer and alpha(n) be the arithmetic function which assigns the multiplicative order of a n modulo n to every integer a coprime to n and vanishes elsewhere. Similarly, let beta(n) assign the projective multiplicative order of a n modulo n to every integer a coprime to n and vanishes elsewhere. In this paper, we present a study of these two arithmetic functions. In particular, we prove that for positive integers n(1) and n(2) with the same square-free part, there exists a relationship between the functions alpha(n1) and alpha(n2) and between the functions beta(n1) and beta(n2). This allows us to reduce the determination of alpha(n) and beta(n) to the case where n is square-free. These arithmetic functions recently appeared in the context of an old problem of Molluzzo, and more precisely in the study of which arithmetic progressions yield a balanced Steinhaus triangle in Z/n Z for n odd.