Divisor Product

Author

Eric W. Weisstein
September 1, 2008

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

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

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

Definition

V6

DivisorProduct[n_Integer]:=Times@@Divisors[n]
DivisorProduct[n_Integer]:=Module[{d},Product[d,{d,Divisors[n]}]]

V7

DivisorProduct[n_, f_:Identity, cond___]:=Exp[DivisorSum[n,Function[x,Log[f[x]]],cond]]

Values

d=Table[DivisorProduct[n],{n,20}]

DivisorProductPower[p_,max_:20]:=Module[{d,n},
d=Table[DivisorProduct[n],{n,max}];
Position[d^(1/p),_Integer,1]//Flatten
]

DivisorProduct/@{1,6,8,10,14}

Table of n such that π(n) is a kth power

Table[Select[Range[200],IntegerQ[DivisorProduct[#]^(1/n)]&,10],{n,5}]//Column

Identity

(∏_(d|n))d==(n^((Subscript[σ, 0])(n)/2))

Union[Table[Product[d,{d,Divisors[n]}]==n^(DivisorSigma[0,n]/2),{n,15}]]

Union[Table[DivisorProduct[n]==n^(DivisorSigma[0,n]/2),{n,15}]]

DivisorProduct[n,#&]

DivisorProduct(n,(((1-(x^(n/(#1)))))^(MoebiusMu[#1]))&)==(Cyclotomic[n, x])

Cyclotomic

DivisorProduct[n,(((1-(x^(n/#))))^(MoebiusMu[#]))&]

{DivisorProduct[n,(((1-(x^(n/#))))^(MoebiusMu[#]))&],Cyclotomic[n,x]}/.n->1

Subtract@@@Table[{(E^(DivisorSum[n,Function[x$,Log[((((1-(x^(n/(#1)))))^(MoebiusMu[#1]))&)[x$]]]])),Cyclotomic[n,x]},{n,2,1000}]//Simplify//Union

DivisorProduct(n,(Cyclotomic[#1, x])&)==(x^n)-1

DivisorProduct[n,Cyclotomic[#,x]&]

Subtract@@@Table[{DivisorProduct[n,Cyclotomic[#,x]&],(x^n)-1},{n,1000}]//Simplify//Union

Product[(Subscript[d, i])]=(n^2)

1+Position[Table[PowerOf[DivisorProduct[n],n],{n,2,100}],2]//Flatten

Product[(Subscript[d, i])]=(n^3)

p1=1+Position[Table[PowerOf[DivisorProduct[n],n],{n,2,2000}],3]//Flatten

p2=Position[Table[Length[Divisors[n]],{n,2000}],6]//Flatten

Union[p1-p2]

Product[(Subscript[d, i])]=(n^4)

PowerOf[p_,x_]:=Module[{r=Log[p]/Log[x]},If[Abs[r-Round[r]]<.00001,Round[r],N[r]]]

p1=1+Position[Table[PowerOf[DivisorProduct[n],n],{n,2,1000}],4]//Flatten

p2=Position[Table[Length[Divisors[n]],{n,1000}],8]//Flatten

Union[p1-p2]

Product[(Subscript[d, i])]=(n^5)

p1=1+Position[Table[PowerOf[DivisorProduct[n],n],{n,2,2000}],5]//Flatten

p2=Position[Table[Length[Divisors[n]],{n,2000}],10]//Flatten

Union[p1-p2]