Abelian Group

Author

Eric W. Weisstein
July 28, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/GroupTheory/AbelianGroup.nb.

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

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

Counts

Package

NumberOfAbelianGroups/@Range[20]

Built-In

FiniteAbelianGroupCount[Range[20]]

FiniteAbelianGroupCount[Range[1024]]==NumberOfAbelianGroups/@Range[1024]

Numbers of Nonisomorphic Abelian Groups of Order 1, 2, …

Timing[a=Table[NumberOfAbelianGroups[n],{n,10^6}];]

Smallest Missing Number of Nonisomorphic Abelian groups

max=Position[Equal@@@Transpose[{u=Union[a],Range[Length[u]]}],False,1,1][[1,1]]

Smallest Orders having 1, 2, 3, … Abelian groups

orders=Rest[First/@Split[FoldList[Max,0,a]]]

Flatten[Position[a,#,1,1]&/@orders]

Sums

(∑_(n=1)^∞)((a(n))/(n^s))=ζ(s) ζ(2 s)⋯

V6

Product[Zeta[n s],{n,∞}]

V7

Product[Zeta[n s],{n,∞}]

(∑_(n=1)^N)a(n)

A063966

Rest[FoldList[Plus,0,Take[a,20]]]

<<Utilities`Plot`

ListPlot[Rest[FoldList[Plus,0,Take[a,1000]]],Joined->True,PlotStyle->Red]

Plot[2.29x-14.65x^(1/2)+118.69x^(1/3),{x,0,1000},PlotStyle->Red]

Table[
Product[Zeta[j/k],{j,k-1}]Product[Zeta[j/k],{j,k+1,∞}],
{k,3}]

N[%,30]

V6

Product[Zeta[j],{j,2,∞}]

V7

Product[Zeta[j],{j,2,∞}]

(Subscript[A, 2])

Zeta[(1/2)]SequenceLimit[N[FoldList[Times,1,Table[Zeta[j/2],{j,3,100}]],200]]//Timing

Zeta[(1/2)]SequenceLimit[N[FoldList[Times,1,Table[Zeta[j/2],{j,3,200}]],100]]//Timing

N[Zeta[1/2]Times@@Table[Zeta[j/2],{j,3,#}],110]&/@{1300,1400,2000}

(Subscript[A, 3])

Zeta[(1/3)] Zeta[(2/3)]SequenceLimit[N[FoldList[Times,1,Table[Zeta[j/3],{j,4,200}]],200]]//Timing

N[Zeta[1/3]Zeta[2/3]Times@@Table[Zeta[j/3],{j,4,#}],110]&/@{1300,1400,2000}

(∑_(n=1)^N)(1/(a(n)))

N[Times@@(1-Sum[(1/PartitionsP[k-1]-1/PartitionsP[k])/#^k,{k,2,200}]&/@Prime[Range[500]]),20]//Timing

N[Times@@(1-Sum[(1/PartitionsP[k-1]-1/PartitionsP[k])/#^k,{k,2,250}]&/@Prime[Range[500]]),20]//Timing

Timing[q[200]=Times@@(1-Sum[(1/PartitionsP[k-1]-1/PartitionsP[k])/#^k,{k,2,200}]&/@Prime[Range[1000]]);]

Timing[q[250]=Times@@(1-Sum[(1/PartitionsP[k-1]-1/PartitionsP[k])/#^k,{k,2,250}]&/@Prime[Range[1000]]);]

Timing[q[300]=Times@@(1-Sum[(1/PartitionsP[k-1]-1/PartitionsP[k])/#^k,{k,2,300}]&/@Prime[Range[1000]]);]

N[{q[200],q[250],q[300]},75]

V7

1-Sum[(1/PartitionsP[k-1]-1/PartitionsP[k])/Prime[n]^k,{k,2,∞}]

Product[%,{n,∞}]