Primorial

Author

Eric W. Weisstein
March 22, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/Primorial.nb.

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

©2009 Wolfram Research, Inc. except for portions noted otherwise

Code

Notation

Primorial[n]//TraditionalForm

Primorial[Prime[n]]//TraditionalForm

Enumeration

Primorial[Range[10]]

Primorial2[Range[10]]

Plot

<<Utilities`Plot`

IntegerListPlot[Log[Primorial/@Range[10]],AxesLabel->TraditionalForm/@{n,Log[pr[n]]}]

Limits

(lim_(n->∞)) ((((Subscript[p, n])#))^(1/(Prime[n])))==e

Calculation

With[{n=10^4},(Times@@N[Prime[Range[n]],10])^(1/Prime[n])]//Timing

With[{n=10^5},(Times@@N[Prime[Range[n]],10])^(1/Prime[n])]//Timing

With[{n=10^5},((Primorial[Prime[n]])^(1./Prime[n]))]//N//Timing

With[{n=10^6},(Times@@N[Prime[Range[n]],10])^(1/Prime[n])]//Timing

With[{n=10^6},((Primorial[Prime[n]])^(1./Prime[n]))]//N//Timing

With[{n=10^7},(Times@@N[Prime[Range[n]],10])^(1/Prime[n])]//Timing

Plot

data=MapIndexed[#1^(1./Prime[#2[[1]]])&,Primorial/@Range[5000]];//Timing

MakeBoxes[ylabel,TraditionalForm]:=SuperscriptBox[RowBox[{"(",RowBox[{SubscriptBox["p","n"]," ","#"}],")"}],RowBox[{"1","/",SubscriptBox["p","n"]}]]

ListPlot[data,
AxesLabel->TraditionalForm/@{n,ylabel},PlotJoined->True,PlotRange->{Automatic,{2.5,2.75}},PlotStyle->Red,Epilog->{Blue,Dashing[{.02}],Line[{{0,E},{5000,E}}]}]

(lim_(n->∞)) ((Prime[n])/((((Prime[n])#))^(1/n)))==?

Jeff Boscole
Thu, 12 Mar 2009 17:22:02 -0500

ListPlot[Table[Module[{p=Prime[n]},(p/((Primorial[p])^(1./n)))],{n,10^4}],Joined->True,PlotStyle->Red]

With[{n=10^4},Module[{p=Prime[n]},(p/((Primorial[p])^(1./n)))]]//Timing

With[{n=10^6},Module[{p=Prime[n]},(p/((Primorial[p])^(1./n)))]]//Timing

With[{n=10^7},Module[{p=Prime[n]},(p/((Primorial[p])^(1./n)))]]//Timing