ON THE DISTRIBUTION OF SQUARE-FULL AND CUBE-FULL NUMBERS

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. Such k-full numbers (k = 2, 3) are considered which are at the same time 1-free (1 greater than or equal to k + 2). At first an asymptotic result is given for the number N-k,N-1(x) of k-full and 1-free numbers not exceeding x. Then the distribution of these numbers in short intervals is investigated. We obtain different estimations of the difference N-k,N-1(x + h) - N-k,N-1(x) in the cases k = 2, 1 = 4, 5, 6, 7, 1 greater than or equal to 8 and k = 3, 1 = 5, 6, 7, 1 greater than or equal to 8.