How often is the number of divisors of n a divisor of n ?
Let d(n) denote the number of positive integers dividing the positive integer n. The author establishes (by ingenious arguments) the following results: Theorem 1. \[ \#\{n\leq x:\quad d(n) | n\}=(x/\sqrt{\log x})(\log \log x)^{-1+o(1)}. \] Theorem 2. Let \(\{\xi_ k\}^{\infty}_{k=0}\) be defined recursively by \(\xi_ 0=0\), \(\xi_ k=\xi_{k-1}+2^{\xi_{k-1}}\) for \(k\geq 1\). Then there is a constant C such that \[ \#\{n\leq x:\quad d(n) | n\}\quad \leq \frac{C x (\log \log \log \log x)^{1/2} \xi (x)}{(\log x)^{1/2} \log \log x} \] for \(x\geq e^{16}\), where \(\xi (x)=\#\{k\geq 0\), \(\xi_ k\leq x\}\).
scientific article; zbMATH DE number 3906585