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On path partitions of the divisor graph

Volume 192 / 2020

Paul Melotti, Éric Saias Acta Arithmetica 192 (2020), 329-339 MSC: Primary 11N37, 11B75; Secondary 05C38, 05C70. DOI: 10.4064/aa180711-26-4 Published online: 25 November 2019

Abstract

It is known that the longest simple path in the divisor graph that uses integers $\leq N$ is of length $\asymp N/\!\log N$. We study the partitions of $\{1,\dots , N\}$ into a \emph {minimal} number of paths in the divisor graph, and we show that in such a partition, the longest path can have length asymptotically $N^{1-o(1)}$.

Authors

  • Paul MelottiSorbonne Université
    LPSM
    4 place Jussieu
    F-75005 Paris, France
    e-mail
  • Éric SaiasSorbonne Université
    LPSM
    4 place Jussieu
    F-75005 Paris, France
    e-mail

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