On the distribution of square-full and cube-full numbers

Abstract

There are many results on the distribution of square-full and cube-full numbers. In this article the distribution of these numbers are studied in more detail. Suchk-full numbers (k=2,3) are considered which are at the same time 1-free (1≥k+2). At first an asymptotic result is given for the numberNk,1(x) ofk-full and 1-free numbers not exceedingx. Then the distribution of these numbers in short intervals is investigated. We obtain different estimations of the differenceNk,1(x+h)−Nk,1(x) in the casesk=2, 1=4,5,6,7,1≥8 andk=3, 1=5,6,7, 1≥8.