In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$ with the definition $\psi(1)=1.$
Definition. Inspired in the book [1] (pages 144-146), I define the Luca-Somer numbers as the positive integers $n\geq 1$ such that $n\mid a^{\psi(n)}-b^{\psi(n)}$ whenever $(a\cdot b,n)=1$.
Claim. It is inmeditate to check that Rotkewicz numbers of the form $n=pq$ that got Rotkiewiccz in his works about a similar question and that present the authors, are also Luca-somer numbers.
An example is $n=2\cdot 3$. On the other hand I've considered that it is easy the proof, in a same way that did the authors, to prove a Lemma that asserts for our definition.
Lemma. A positive integer $n\geq 1$ is a Luca-Somer number if an only if $c^{\psi(n)}\equiv 1\text{ mod }n$, whenever the integer $c\geq 1$ satisfies $(c,n)=1$.
And from here I wondered about more examples of families (maybe exploting as did the authors some alternative characterization of Lucas-Somer numbers, as they computed by using their Lemma 12.31).
Question. I would like to know if you can provide reasonings to get more families of Luca-Somer numbers (maybe a different characterization of our Definition is required). Many thanks.
The authors compute such families of Rotkiewicz numbers as Theorem 12.32, 12.33 and 12.34 exploiting their characterization of Rotkiewicz numbersProposition 12.32.
References:
[1] Michael Křižek, Florian Luca and Lawrence Somer, 17 Lectures on Fermat Numbers, CMS Books in Mathematics, Canadian Mathematical Society, Springer-Verlag (2001).
