A counting function for the sequence of perfect powers

A natural number of the form mn where m is a positive integer and n 2 is called a perfect power. Unsolved problems concerning the set of perfect powers abound throughout much of number theory. The most famous of these is known as the Catalan conjecture, which states that the only perfect powers which differ by unity are the integers 8 and 9. It is of interest to note that this particular problem has only recently been solved using rather deep results from the theory of cyclotomic fields (see [4]). The set of perfect powers can naturally be arranged into an increasing sequence of distinct integers, in which those perfect powers expressible with different exponents are treated as a single element of the sequence.