Cube-full primitive roots

Adopting the methods of Cohen–Davis [E. Cohen and K. J. Davis, Elementary estimates for certain types of integers, Acta Sci. Math. (Szeged) 31 (1970) 363–371] and Suryanarayana–Subrahmanyam [D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math. 12 (1981) 175–190], an elementary method to derive an asymptotic estimate for the number of primitive roots modulo a prime that are cube-full numbers is presented.