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I have read that for the Landau function $g(n)$ (http://mathworld.wolfram.com/LandausFunction.html), one knows that

$\lim_{n \rightarrow \infty} g(n+1)/g(n) = 1$

(stated, but not proved in "On Landau's function $g(n)$" of Jean-Louis Nicolas. I tried to find it in the French source that he gave, but it is rather long and my French is rather non-existing.)

I wonder whether someone has derived an explicit constant bound in the sense that

$g(n+1) \leq C g(n)$ for $n \geq N$

where $C$ and $N$ can be given explicitly (and $N$ is "small").

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    $\begingroup$ How about $g(n+1)\le 2g(n)$ for all $n$. $\endgroup$
    – Anthony Quas
    Commented Jul 1, 2015 at 14:39
  • $\begingroup$ Can you sketch me the argument? Thanks. $\endgroup$
    – J Fabian Meier
    Commented Jul 1, 2015 at 19:41
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    $\begingroup$ What I had in mind is that $g(n)$ is the largest value of $p_1^{\alpha_1}\times\dots \times p_j^{\alpha_j}$ such that $p_1^{\alpha_1}+\ldots+p_j^{\alpha_j}\le n$. For all values of $n$ that are at least 4, this is obtained with one of the $p_j$'s being equal to 2. Now if $g(n+1)=2^{\alpha_1}\times\dots \times p_j^{\alpha_j}$ with $2^{\alpha_1}+\ldots+p_j^{\alpha_j}\le n+1$, then $2^{\alpha_1-1}+p_2^{\alpha_2}+\ldots+p_j^{\alpha_j}\le n$ and $g(n)\ge 2^{\alpha_1-1}\times p_2^{\alpha_2}\times\dots\times p_j^{\alpha_j}=g(n+1)/2$. $\endgroup$
    – Anthony Quas
    Commented Jul 1, 2015 at 19:49
  • $\begingroup$ Nice and sharp. Once you find out how big a power of two is needed, say $c$ many, you can later establish something like $g(n) \geq g(n + 2^{c-1})/2$. Gerhard "Perhaps Already In The Literature" Paseman, 2015.07.01 $\endgroup$
    – Gerhard Paseman
    Commented Jul 1, 2015 at 19:54
  • $\begingroup$ @AnthonyQuas Actually $g(n)$ is even except for $n=3, 8, 15$. This is proved e.g. in Nicolas, Jean-Louis Ordre maximal d'un élément du groupe Sn des permutations et "highly composite numbers''. (French) Bull. Soc. Math. France 97 1969 129–191 (available online, numdam.org/article/BSMF_1969__97__129_0.pdf p142). The fact that $g(n+1) \leq 2 g(n)$ is the Corollary p143. $\endgroup$
    – Gagar
    Commented Feb 9, 2019 at 21:19

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