Résumé
Journal of Integer Sequences 13, article 10.2.1, 2010 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 an exact relationship between the functions$\alpha_{n_1}$and$\alpha_{n_2}$and between the functions$\beta_{n_1}$and$\beta_{n_2}$ . 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$\mathbb{Z}/n\mathbb{Z}$for$n$odd.