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

2,1

Comments

Massias proved that the function log(g(n))/sqrt(n*log(n)) reaches its maximum at n = 1319766. Therefore this sequence is finite, with a(378) = 1319766 being the last term. - Amiram Eldar, Aug 23 2019

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

Example

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)

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 *)

Crossrefs

Cf. A000793.

Sequence in context: A047371 A327492 A044918 * A133677 A362403 A075750

Adjacent sequences: A103632 A103633 A103634 * A103636 A103637 A103638

Keyword

nonn,fini,full

Author

Eric W. Weisstein, Feb 11 2005

Extensions

More terms from R. J. Mathar, Feb 14 2008

More terms from Alois P. Heinz, Feb 18 2013

Status

approved