Grantham, Jon The largest prime dividing the maximal order of an element of \(S_ n\). (English) Zbl 0824.11059 Math. Comput. 64, No. 209, 407-410 (1995). Let \(g(n)\) denote the maximal order of an element of the symmetric group on \(n\) letters. E. Landau [Arch. Math. Phys., III. Ser. 5, 92-103 (1903; F.d.M. 34, 233)] proved that \(\log g(n)\sim (n\log n)^{1/2}\) as \(n\to \infty\). For improved estimates see J.-P. Massias, J.-L. Nicolas and G. Robin [Acta Arith. 50, 221-242 (1988; Zbl 0646.10037)]. Let \(P(g (n))\) be the largest prime divisor of \(g(n)\). J.-L. Nicolas [Acta Arith. 14, 315-332 (1968; Zbl 0179.348)] showed that \(P(g (n))\sim (n\log n)^{1/2}\) as \(n\to \infty\). J.-P. Massias, J.-L. Nicolas and G. Robin [Math. Comput. 53, 665-678 (1989; Zbl 0675.10028)] obtained that \(P(g (n))\leq 2.86 (n\log n)^{1/2}\) for \(n\geq 2\). They conjectured that \(P(g (n)) (n\log n)^{-1/2}\) achieves a maximum \((1.265 \ldots)\) for \(n\geq 5\) at \(n=215\). In the paper under review the author proves that \(P(g (n))\leq 1.328 (n\log n)^{1/2}\) for \(n\geq 5\). Reviewer: M.Szalay (Budapest) Cited in 5 Documents MSC: 11N45 Asymptotic results on counting functions for algebraic and topological structures 20B40 Computational methods (permutation groups) (MSC2010) 20B05 General theory for finite permutation groups Keywords:maximal order in the symmetric group on \(n\) letters Citations:Zbl 0646.10037; Zbl 0179.348; Zbl 0675.10028 PDFBibTeX XMLCite \textit{J. Grantham}, Math. Comput. 64, No. 209, 407--410 (1995; Zbl 0824.11059) Full Text: DOI arXiv Online Encyclopedia of Integer Sequences: Landau’s function g(n): largest order of permutation of n elements. Equivalently, largest LCM of partitions of n. For the Landau function L(n), A000793, this sequence gives the largest prime which is a factor of L(n).