On the $\varkappa\,$th root of a Stieltjes moment sequence

Stieltjes moment sequences $\{a_n\}_{n=0}^\infty$ whose $\varkappa\,$th roots $\{\sqrtk{a_n}\}_{n=0}^\infty$ are Stieltjes moment sequences are studied ($\varkappa$ is a fixed integer greater than or equal to 2). A formula connecting the closed supports of representing measures of $\{a_n\}_{n=0}^\infty$ and $\{\sqrtk{a_n}\}_{n=0}^\infty$ is established. The relationship between the holes of the supports of these measures is investigated. The set of all pairs $(M,N)$ of positive integers for which there exists a Stieltjes moment sequence whose square root is a Stieltjes moment sequence and both of them have representing measures supported on subsets of $(0,\infty)$ of cardinality $M$ and $N$, respectively, is described.