Let n be a positive integer and α_n be the arithmetic function which assigns the multiplicative order of aⁿ modulo n to every integer a coprime to n and vanishes elsewhere. Similarly, let β_n assign the projective multiplicative order of aⁿ 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 n₁ and n₂ with the same square-free part, there exists a relationship between the functions α_n1 and α_n2 and between the functions β_n1 and β_n2. This allows us to reduce the determination of α_n and β_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/nZ for n odd.