In this work, we study eigenvalue distribution results of a class of highly non-normal matrix-sequences, which can be viewed as a low rank perturbation depending on a parameter $\beta>1$ of the basic Toeplitz matrix-sequence $\{T_n(e^{\mathbf{i}\theta})\}_{n\in\N}$, $\mathbf{i}^2=-1$. The latter has obviously all eigenvalues equal to zero for any matrix order $n$, while for the matrix-sequence under consideration we will show a strong clustering at the range of the generating function $e^{\mathbf{i}\theta}$ i.e. at the complex unit circle. For $\beta\ge 2$ no outliers show up, while for $\beta \in (1,2)$ only two outliers are present, which are both real, positive and have finite limits equal to $\beta-1$ and $(\beta-1)^{-1}$, respectively. The problem looks mathematically innocent, but indeed it is quite challenging since all the sophisticated machinery for deducing the eigenvalue clustering is not easy to apply in the current setting and at most we may hope for weak clustering results. In the derivations, we resort to a trick already used for the spectral analysis of the Google matrix plus several tools from complex analysis. We only mention that the problem is not an academical curiosity and in fact it stems from problems in dynamical systems and number theory. Numerical experiments in high precision are provided and more results are sketched for limit case of $\beta=1$.