Abstract
An integer n is called lexicographic if the increasing sequence of its divisors, regarded as words on the (finite) alphabet of the prime factors (arranged in increasing size), is ordered lexicographically. This concept easily yields to a new type of multiplicative structure for the exceptional set in the Maier-Tenenbaum theorem on the propinquity of divisors, which settled a well-known conjecture of Erdös. We provide asymptotic formulae for the number of lexicographic integers not exceeding a given limit, as well as for certain arithmetically weighted sums over the same set. These results are subsequently applied to establishing an Erdös-Kac theorem with remainder for the distribution of the number of prime factors over lexicographic integers. This provides quantitative estimates for lexicographical exceptions to Erdos' conjecture that also satisfy the Hardy-Ramanujan theorem.
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Stef, A., Tenenbaum, G. Entiers Lexicographiques. The Ramanujan Journal 2, 167–184 (1998). https://doi.org/10.1023/A:1009770126879
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DOI: https://doi.org/10.1023/A:1009770126879