Classification of cyclic groups underlying only smooth skew morphisms

A skew morphism of a finite group A is a permutation \(\varphi \) $\phi$ of A fixing the identity element and for which there is an integer-valued function \(\pi \) $\pi$ on A such that \(\varphi (ab)=\varphi (a)\varphi ^{\pi (a)}(b)\) $\phi (ab)=\phi (a){\phi }^{\pi (a)}(b)$ for all \(a, b \in A\) $a,b\in A$ . A skew morphism \(\varphi \) $\phi$ of A is smooth if the associated power function \(\pi \) $\pi$ is constant on the orbits of \(\varphi \) $\phi$ , that is, \(\pi (\varphi (a))\equiv \pi (a)\pmod {|\varphi |}\) $\pi (\phi (a))\equiv \pi (a)(mod|\phi |)$ for all \(a\in A\) $a\in A$ . In this paper, we show that every skew morphism of a cyclic group of order n is smooth if and only if \(n=2^en_1\) $n=2^e{n}_1$ , where \(0 \le e \le 4\) $0\le e\le 4$ and \(n_1\) $n_1$ is an odd square-free number. A partial solution to a similar problem on non-cyclic abelian groups is also given.