Counting Formulas for Exponentially Odd and Exponentially Even Integers

Abstract : A positive integer is said to be exponentially even if its prime factorization contains only even prime powers, and is said to be exponentially odd if its prime factorization contains only odd prime powers. Counting formulas are proved for the number of positive integers (i) which are relatively prime to an exponentially even number and (ii) which are relatively prime to an exponentially odd number. Asymptotic estimates of their corresponding counting functions are also derived.