Let \(f(n)\) denote the number of unordered factorizations of a positive integer \(n\) as a product of factors \(>1\). The authors prove that the number of distinct values \(\leq x\) of \(f(n)\) is \(\leq\exp(9(\log x)^{2/3})\) for all \(x\geq 1\). This improves the first result in [Acta Arith. 142, No. 1, 41–50 (2010;
Zbl 1213.11020)] by
F. Luca,
A. Mukhopadhyay and
K. Srinivas. A key step in the proof is to show that the number \(q(m)\) of representations \(m=\lfloor\sqrt{\alpha_1}\rfloor+ \lfloor\sqrt{\alpha_2}\rfloor+\cdots+ \lfloor\sqrt{\alpha_k}\rfloor\) with positive integers \(\alpha_1\geq \alpha_2\geq\cdots\geq \alpha_k\) satisfies \(q(m)\leq\exp(5m^{2/3})\) for all \(m\geq 1\).
