Silverman Constant

Author

Eric W. Weisstein
April 16, 2004

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

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

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

Eric W. Weisstein
February 1, 2002

As Sum

Plot

<<Utilities`Plot`

IntegerListPlot[Table[1/(EulerPhi[n]DivisorSigma[1,n]),{n,100}]]

ListPlot[Table[1/(EulerPhi[n]DivisorSigma[1,n]),{n,10^3,10^4}],PlotStyle->Red,PlotJoined->True]

Numeric

N[
Plus@@Table[1/(EulerPhi[n]DivisorSigma[1,n]),{n,10^4}],
20]//Timing

one=SetPrecision[1,20];
Plus@@Table[one/(EulerPhi[n]DivisorSigma[1,n]),{n,10^5}]//Timing

one=SetPrecision[1,20];
Plus@@Table[one/(EulerPhi[n]DivisorSigma[1,n]),{n,5 10^5}]//Timing

one=SetPrecision[1,20];
Plus@@Table[one/(EulerPhi[n]DivisorSigma[1,n]),{n,10^6}]//Timing

one=SetPrecision[1,20];
Plus@@Table[one/(EulerPhi[n]DivisorSigma[1,n]),{n,10^7}]//Timing

SequenceLimit[FoldList[Plus,SetPrecision[0,20],Table[1/(EulerPhi[n]DivisorSigma[1,n]),{n,30}]]]//Timing

V7

Sum[1/(EulerPhi[n]DivisorSigma[1,n]),{n,∞}]

As Product/Sum Over Primes

N[Times@@((1+Plus@@Table[(1/(((#)^(2k))-((#)^(k-1)))),{k,100}])&/@Prime[Range[50]]),20]//Timing