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Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:

Can we prove that $\lambda(2^n - 1) \ll 2^{o(n)}$ for infinitely many positive integers $n$?

Thanks for any advice/reference.

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  • $\begingroup$ There might be something in Kazimierz Szymiczek, On the distribution of prime factors of Mersenne numbers, Prace Mat. 13 (1969) 33–49, MR0252316 (40 #5537). $\endgroup$
    – Gerry Myerson
    Commented Jul 26, 2016 at 12:28
  • $\begingroup$ @GerryMyerson It might be, but I didn't found that paper anywhere. $\endgroup$
    – user40023
    Commented Jul 26, 2016 at 13:25
  • $\begingroup$ Can you provide a table of values? I do not have a lambda routine handy. Gerhard "Nor A Sheepda Routine Either" Paseman, 2017.06.06. $\endgroup$
    – Gerhard Paseman
    Commented Jun 6, 2017 at 21:40
  • $\begingroup$ @GerhardPaseman Here jpst.it/-jHl you find all pairs $n, \; \lambda(2^n - 1)$ for $n=1,\ldots,100$ $\endgroup$
    – user40023
    Commented Jun 7, 2017 at 14:45
  • $\begingroup$ The Szymiczek paper is available at wydawnictwa.ptm.org.pl/index.php/commentationes-mathematicae/… $\endgroup$
    – Gerry Myerson
    Commented Sep 15, 2022 at 7:48

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