A remark on a paper of F. Luca and A. Sankaranarayanan (Q616541)

Recall that a positive integer \(n\) is squarefull if \(p^2\) divides \(n\) for all prime factors \(p\) of \(n\). In the paper under review, the author proves that if \(f(n)\) is an integer-valued multiplicative function satisfying certain conditions, then the set of \(n\) such that the summatory function \(f(1)+\dots+f(n)\) is squarefull is of asymptotic density zero. The theorem applies to the Euler function \(\phi(n)\) and to the sum of divisors function \(\sigma(n)\). Thus, the main result of this paper generalizes and extends the main result of the paper [\textit{F. Luca} and \textit{A. Sankaranarayanan}, ``On positive integers \(n\) such that \(\phi(1)+\dots +\phi(n)\) is a square'', Bol. Soc. Mat. Mex., III. Ser. 14, No. 1, 1--6 (2008; Zbl 1211.11005)] referred to in the title, where it was only shown that the set of positive integers \(n\) such that \(\phi(1)+\dots+\phi(n)\) is a square is of asymptotic density zero. After some technical preliminaries, the core of the proof of the main result is similar to the one of Luca and Sankaranarayanan.