On a problem due to Erdős concerning $w_1(n)$

Abstract

Adopting notation from Erdős [Congress. Numer., 1980], we define $w_1(n) = w(n)$ for positive integers $n$ so that $w(1) = 0$ and so that: If $n > 1$ and $n = \prod p^{α_p}$, then $w(n) = \sum_{p^{α_p} \parallel n} \frac{n}{p^{α_p}}$. By then defining $V(x)=\bigl|\{w(n):1\le n\le x\}\bigr|$ for $x \geq 1$, Erdős considered whether or not $V(x) = o(x)$ and left this as an open problem. This same problem was also given as an open problem in Guy's famous textbook on unsolved problems in number theory and seems to have remained open. We solve this problem by proving that $V(x) \neq o(x)$, and, moreover, by proving the stronger result such that $\liminf_{x \to \infty} \frac{V(x)}{x} > 0$.