Backhouse's Constant

Author

Eric W. Weisstein
April 16, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/BackhousesConstant.nb.

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

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

P and Q

BackhouseP[x_?NumericQ,n_:10^3]:=1+Sum[Prime[k]x^k,{k,n}]

BackhouseQ[x_]:=1/BackhouseP[x]

Root of P

Timing[asymp=x/.FindRoot[BackhouseP[x],{x,-.75,-.5},WorkingPrecision->20]]

Timing[asymp=x/.FindRoot[BackhouseP[x],{x,-.75,-.5},WorkingPrecision->100]]

Timing[asymp=x/.FindRoot[BackhouseP[x,10^4],{x,-.75,-.5},WorkingPrecision->100]]

ContinuedFraction[-asymp]

ContinuedFraction[1.456074948582689671399595351116]

1/asymp

Plots

<<Plot`

Show[GraphicsArray[Block[{$DisplayFunction=Identity},
{
Plot[BackhouseP[x],{x,-.99,.75},PlotStyle->Red,
PlotRange->{-10,20},AxesLabel->TraditionalForm/@{x,P[x]}],
PlotDiscontinuous[BackhouseQ[x],{x,-.99,asymp,.75},
VerticalAsymptotes->{asymp},PlotStyle->Red,
PlotRange->{All,{-20,20}},AxesLabel->TraditionalForm/@{x,Q[x]}]
}
]]]

Coefficients

Series[(1/(1+Plus@@Table[Prime[k]x^k,{k,500}])),{x,0,20}]

qs=CoefficientList[Series[(1/(1+Plus@@Table[Prime[k]x^k,{k,2000}])),{x,0,100}],x]

ratio=1/Divide@@@Partition[Abs[qs],2,1]//N

IntegerListPlot[ratio,AxesLabel->TraditionalForm/@{n,Abs[(Subscript[q, n+1])/(Subscript[q, n])]},
Prolog->{Blue,Line[{{0,#},{100,#}}]&@1.456074948582689671399595351116}]