On the number of special numbers

For lack of a better word, a number is called special if it has mutually distinct exponents in its canonical prime factorizaton for all exponents. Let V ( x ) be the number of special numbers ≤ x . We will prove that there is a constant c >1 such that V ( x ) ∼ cx log x . We will make some remarks on determining the error term at the end. Using the explicit abc conjecture, we will study the existence of 23 consecutive special integers.