On the multiplicative order of $a^n$ modulo $n$

Let $n$ be a positive integer and $α_n$ 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 $β_n$ 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 $α_{n_1}$ and $α_{n_2}$ and between the functions $β_{n_1}$ and $β_{n_2}$. 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 $\mathbb{Z}/n\mathbb{Z}$ for $n$ odd.

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