On Some Generalisations of Gauss Sequences
In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler--Gauss sequences. Noting that every Gauss sequence is an Euler--Gauss sequence, we compare them with certain generalisations of Gauss sequences and provide several counterexamples. Unlike Gauss sequences, Euler--Gauss sequences include sequences based on distinct prime factors, such as the Smallest Prime Factor and Greatest Prime Factor sequences (suitably defined at $1$). Moreover, we show that the prime-divisor subclass of Gauss sequences, given by $\sum_{p\mid n} pg_p$ for an integer sequence $(g_n)$, admits a natural extension to Euler--Gauss sequences of the form $\sum_{p\mid n} pf_p(\operatorname{rad}(n))$, where, for each prime $p$, $f_p$ is an integer-valued function and $\operatorname{rad}(n)$ denotes the square-free kernel of $n$. Further, we obtain $q$-analogs of the Euler--Gauss sequences, fill gaps in the literature on $q$-Gauss sequences, and conjecture a divisibility criterion for $q$-Euler--Gauss sequences, which we have verified computationally. We also show that not only do our $q$-Euler--Gauss sequences satisfy the Cyclic Sieving Phenomenon (CSP) exhibited by the $q$-Gauss sequences, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature.