Lexicographic integers
An integer \(n\) is lexicographic if the sequence of its divisors ordered lexicographically (according to the exponents of the increasing sequence of the prime factors of \(n)\) is itself an increasing sequence. Let \(n=\prod^{ \omega (n)}_{j=1} p_j^{\nu_j}\) \((p_1<p_2<\cdots<p_{\omega(n)})\), \(n_j= \prod_{1 \leq i<j} p_i^{\nu_i}\) for \(1\leq j\leq \omega(n)\), \(W(n)=\min_{1\leq j\leq \omega (n)} p_j/n_j \), and \(d_1,\dots,d_{\tau(n)}\) denote the divisors of \(n\) in increasing order. Theorem 1 establishes that \(n\) is lexicographic if and only if \[ W(n)= \min_{1\leq j<\tau(n)}d_{j+1}/d_j. \] \textit{G. Tenenbaum} showed in [Ann. Sci. Éc. Norm. Supér. (4) 19, 1-30 (1986; Zbl 0599.10037)] that the corresponding result with min replaced by max on both sides holds for all \(n\). An asymptotic formula for the number of lexicographic integers \(n\leq x\) is provided as a special case of theorem 2 where a more general weighted sum is considered. The authors go on to establish an Erdős-Kac type of result.
scientific article; zbMATH DE number 1203261