Landau's Function

Author

Eric W. Weisstein
February 11, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/NumberTheoreticFunctions/LandausFunction.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/LandausFunction.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Initialization

<<DiscreteMath`Combinatorica`

Enumeration

LCM

Table[Select[p=Partitions[n],Max[LCM@@@p]==LCM@@#&],{n,15}]//ColumnForm

Max[LCM@@@Partitions[#]]&/@Range[55]//Timing

Max[LCM@@@Partitions[#]]&/@Range[60]//Timing

Max[LCM@@@Partitions[#]]&/@Range[65]//Timing

Max[LCM@@@Partitions[#]]&/@Range[75]//Timing

IntegerListPlot[Log[{1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,210,210,420,420,420,420,840,840,1260,1260,1540,2310,2520,4620,4620,5460,5460,9240,9240,13860,13860,16380,16380,27720,30030,32760,60060,60060,60060,60060,120120,120120,180180,180180,180180,180180,360360,360360,360360,360360,471240,510510,556920,1021020,1021020,1141140,1141140,2042040,2042040,3063060,3063060,3423420,3423420,6126120,6126120,6846840,6846840,6846840,6846840}],AxesLabel->TraditionalForm/@{n,ln[g[n]]}]

SymmetricGroup

GroupTimes[m__List]:=Fold[Permute,First[{m}],Rest[{m}]]

Table[Max[Function[g,Length@NestWhileList[GroupTimes[g,#]&,g,UnsameQ[#,Range[n]]&]]/@SymmetricGroup[n]],{n,8}]//Timing

Asymptotics

Length[a={1,2,3,4,6,6,12,15,20,30,30,60,60,84,105,140,210,210,420,420,420,420,840,840,1260,1260,1540,2310,2520,4620,4620,5460,5460,9240,9240,13860,13860,16380,16380,27720,30030,32760,60060,60060,60060,60060,120120,120120,180180,180180,180180,180180,360360,360360,360360,360360,471240,510510,556920,1021020,1021020,1141140,1141140,2042040,2042040,3063060,3063060,3423420,3423420,6126120,6126120,6846840,6846840,6846840,6846840}]

aratio=MapIndexed[((Log[#1])/(Sqrt[#2[[1]] Log[#2[[1]]]]))&,a]//N;

<<Utilities`Plot`

MakeBoxes[ylabel,TraditionalForm]:=FractionBox[RowBox[{"ln","(",RowBox[{"g","(","n",")"}],")"}],SqrtBox[RowBox[{"n"," ","ln"," ","n"}]]]

IntegerListPlot[aratio,AxesLabel->TraditionalForm/@{n,ylabel}]

Last[HighWaterMarks[Rest[aratio]]]+1

Greatest Prime Factor

GreatestPrimeFactor/@a

MakeBoxes[label,TraditionalForm]:=RowBox[{RowBox[{"gpf","(",RowBox[{"g","(","n",")"}],")"}],"/",SqrtBox[RowBox[{"n"," ","ln"," ","n"}]]}]

IntegerListPlot[MapIndexed[GreatestPrimeFactor[#1]/Sqrt[#2[[1]]Log[#2[[1]]]]&,a],AxesLabel->TraditionalForm/@{n,label}]