Odd Divisor Function

Author

Eric W. Weisstein
September 1, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Sums

DivisorSum[n,#1&,OddQ[#1]&]==DivisorSum[n,(((((-1))^(#1+1)) n)/(#1))&]==(DivisorSigma[1, n])-2 If[OddQ[n],0,(DivisorSigma[1, (n/2)])]

DivisorSum[n,#&,OddQ[#]&]

Subtract@@@Table[{DivisorSum[n,#&,OddQ[#]&],DivisorSum[n,(((-1))^(#+1))(n/(#))&]},{n,100}]//Union

FullSimplify[DivisorSum[n,#&,OddQ[#]&]-DivisorSum[n,(((-1))^(#+1))(n/(#))&],n∈Integers&&n>0]

DivisorSum[n,(((-1))^(#+1))(n/(#))&]

Subtract@@@Table[{DivisorSum[n,#&,OddQ[#]&],DivisorSigma[1,n]-2If[OddQ[n],0,DivisorSigma[1,n/2]]},{n,1000}]//Union

Generating Function

Sum[((n (x^n))/(1+(x^n))),{n,0,∞}]//InputForm

Series[Sum[((n (x^n))/(1+(x^n))),{n,0,100}],{x,0,15}]

Series[(1/24)(((EllipticTheta[3,0,x])^4)+((EllipticTheta[2,0,x])^4)-1),{x,0,15}]

SeriesCoefficient[(1/24)(((EllipticTheta[3,0,x])^4)+((EllipticTheta[2,0,x])^4)-1),{x,0,n}]

Table[{DivisorSum[n, #1 & , OddQ[#1] & ], DivisorSum[n, 
     ((-1)^(#1 + 1)*n)/#1 & ], DivisorSigma[1, n] - 
     2*If[OddQ[n], 0, DivisorSigma[1, n/2]]}*x^n, {n, 10}]

GeneratingFunction[DivisorSum[n, #1 & , OddQ[#1] & ],n,x]