On the distribution of \((k,r)\)-integer in an arithmetic progression (Q2062930)

Let \(r\) and \(k\) be integers such that \(1<r<k\). A positive integer \(n\) is called a \((k,r)\)-integer if it can be written in the form \(n=a^kb\) with integers \(a,b\) such that \(b\) is \(r\)-free (that is, not divisible by the \(r\)-th power of a prime). Using the method of exponent pairs, the author proves an asymptotic formula for the quantity \[ Q_{k,r}(x;l,q) = \bigl\{n\leq x:\mbox{\(n\) is an \((k,r)\)-integer and }n\equiv l\bmod q\}, \] improving on a result by \textit{E. Brinitzer} [Monatsh. Math. 80, 31--35 (1975; Zbl 0318.10028)].