We consider Artin’s conjecture on primitive roots and related Artin-type problems, working over a number field K. Such problems deal with the reductions of algebraic numbers \(\alpha \in K^\times \) modulo primes \(\mathfrak p\) of K. The key property concerns the value of the index of \((\alpha \ \textrm{mod}\ \mathfrak p)\) , but it is also customary to require an additional Frobenius condition. The set of primes \(\mathfrak p\) satisfying such properties admits a density, conditionally under the generalized Riemann hypothesis. In this work, we compare the density for \(\alpha \) to the density for its powers, and also address some related questions.