  We prove several results regarding the distribution of numbers that are the
product of a prime and a $k$-th power. First, we prove an asymptotic formula
for the counting function of such numbers; this generalises a result of E.
Cohen. We then show that the error term in this formula can be sharpened on the
assumption of the Riemann hypothesis. Finally, we prove an asymptotic formula
for these counting functions in short intervals.
