A103635 Positions of running maxima of log(g(n))/sqrt(n*log(n)), where g(n) is Landau's function A000793.
2, 3, 5, 7, 9, 10, 12, 17, 19, 30, 36, 40, 43, 47, 49, 53, 60, 64, 66, 70, 83, 85, 89, 108, 112, 141, 172, 209, 250, 258, 293, 301, 321, 340, 348, 360, 368, 401, 413, 421, 480, 533, 541, 608, 626, 679, 697, 752, 770, 831, 849, 914, 932, 1021, 1118, 1160, 1219
Offset: 2
Examples
From _Jon E. Schoenfield_, Aug 16 2015: (Start)
Terms are the values of n at which record high values of the ratio log(g(n))/sqrt(n*log(n)) occur (where g(n) = A000793(n)):
n g(n) log(g(n))/sqrt(n*log(n))
== ==== ========================
1 1 (undefined)
a(1) = 2 2 0.588705 <--- record high
a(2) = 3 3 0.605148 <--- record high
4 4 0.588705
a(3) = 5 6 0.631623 <--- record high
6 6 0.546467
a(4) = 7 12 0.673286 <--- record high
8 15 0.663955
a(5) = 9 20 0.673666 <--- record high
a(6) = 10 30 0.708800 <--- record high
(End)
Links
- Amiram Eldar, Table of n, a(n) for n = 2..378 (calculated using the MAPLE code by Deléglise et al.; terms 2..123 from Alois P. Heinz)
- Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Landau's function for one million billions, Journal de Théorie des Nombres de Bordeaux, Vol. 20, No. 3 (2008), pp. 625-671.
- Marc Deléglise, Jean-Louis Nicolas, and Paul Zimmermann, Computation of the Landau function g(n) (MAPLE code).
- Jean-Pierre Massias, Majoration explicite de l'ordre maximum d'un élément du groupe symétrique, Annales de la Faculté des sciences de Toulouse: Mathématiques, Vol. 6, No. 3-4 (1984), pp. 269-281.
- Eric Weisstein's World of Mathematics, Landau's Function
Crossrefs
Cf. A000793.
Programs
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Mathematica
g[n_] := Max@Apply[LCM, IntegerPartitions@n, 1]; f[n_] := Log[g[n]]/Sqrt[n * Log[n]]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 2, 100}]; s (* Amiram Eldar, Aug 23 2019 after Robert G. Wilson v at A000793 *)
Extensions
More terms from R. J. Mathar, Feb 14 2008
More terms from Alois P. Heinz, Feb 18 2013
Comments