Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 JordanTotient

Source Notebook

Evaluate Jordan's totient function

Contributed by: Jan Mangaldan

ResourceFunction["JordanTotient"][k,n]

gives the Jordan totient function Jk(n).

Details

Integer mathematical function, suitable for both symbolic and numerical manipulation.
The Jordan totient function Jk(n) gives the number of k-tuples of positive integers that are less than or equal to n that form a coprime (k+1)-tuple together with n.
For a number with u a unit and pi primes, ResourceFunction["JordanTotient"][k,n] gives .
ResourceFunction["JordanTotient"][1,n] is equivalent to EulerPhi[n].
ResourceFunction["JordanTotient"] automatically threads over lists.

Examples

Basic Examples (2) 

Evaluate J1(10):

In[1]:=
ResourceFunction["JordanTotient"][1, 10]
Out[1]=

Plot JordanTotient with log-scaled values:

In[2]:=
(* Evaluate this cell to get the example input *) CloudGet["https://www.wolframcloud.com/obj/57711127-5bb2-4af3-9b65-142cf9a9804c"]
Out[2]=

Scope (2) 

Show a table of Jordan totients:

In[3]:=
TableForm[Array[ResourceFunction["JordanTotient"], {7, 7}], TableHeadings -> {Range[7], Range[7]}]
Out[3]=

JordanTotient threads elementwise over lists:

In[4]:=
ResourceFunction["JordanTotient"][3, {2, 4, 6}]
Out[4]=

Applications (2) 

Verify Gegenbauer's formula:

In[5]:=
With[{k = 5, l = 7, n = 31}, ResourceFunction["JordanTotient"][k + l, n] == DirichletConvolve[d^k ResourceFunction["JordanTotient"][l, d], ResourceFunction["JordanTotient"][k, d], d, n]]
Out[5]=

A formula for the logarithmic derivative of a cyclotomic polynomial evaluated at 1 due to Lehmer:

In[6]:=
With[{k = 5, n = 17}, Limit[D[Log[Cyclotomic[n, t]], {t, k}], t -> 1] == \!\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(k\)]\( \*FractionBox[\(BernoulliB[j, 1] StirlingS1[k, j]\), \(j\)] \* InterpretationBox[ TagBox[ DynamicModuleBox[{Typeset`open = False}, FrameBox[ PaneSelectorBox[{False->GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["JordanTotient", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"], ImageSizeCache->{11., {1., 10.}}], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"], ImageSizeCache->{11., {1., 10.}}]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}], True->GridBox[{ {GridBox[{ { PaneBox[GridBox[{ { StyleBox[ StyleBox[ AdjustmentBox["\<\"[\[FilledSmallSquare]]\"\>", BoxBaselineShift->-0.25, BoxMargins->{{0, 0}, {-1, -1}}], "ResourceFunctionIcon", FontColor->RGBColor[ 0.8745098039215686, 0.2784313725490196, 0.03137254901960784]], ShowStringCharacters->False, FontFamily->"Source Sans Pro Black", FontSize->0.6538461538461539 Inherited, FontWeight->"Heavy", PrivateFontOptions->{"OperatorSubstitution"->False}], StyleBox[ RowBox[{ StyleBox["JordanTotient", "ResourceFunctionLabel"], " "}], ShowAutoStyles->False, ShowStringCharacters->False, FontSize->Rational[12, 13] Inherited, FontColor->GrayLevel[0.1]]} }, GridBoxSpacings->{"Columns" -> {{0.25}}}], Alignment->Left, BaseStyle->{LineSpacing -> {0, 0}, LineBreakWithin -> False}, BaselinePosition->Baseline, FrameMargins->{{3, 0}, {0, 0}}], ItemBox[ PaneBox[ TogglerBox[Dynamic[Typeset`open], {True-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeCloser"]], False-> DynamicBox[FEPrivate`FrontEndResource[ "FEBitmaps", "IconizeOpener"]]}, Appearance->None, BaselinePosition->Baseline, ContentPadding->False, FrameMargins->0], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{1, 1}, {0, 0}}], Frame->{{ RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627, 0.5], False}, {False, False}}]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}, GridBoxSpacings->{"Columns" -> {{0}}, "Rows" -> {{0}}}]}, { StyleBox[ PaneBox[GridBox[{ { RowBox[{ TagBox["\<\"Version (latest): \"\>", "IconizedLabel"], " ", TagBox["\<\"1.0.0\"\>", "IconizedItem"]}]}, { TagBox[ TemplateBox[{"\"Documentation »\"", "https://resources.wolframcloud.com/FunctionRepository/resources/055b08ea-79a0-458a-8254-4720bc316c4f/"}, "HyperlinkURL"], "IconizedItem"]} }, DefaultBaseStyle->"Column", GridBoxAlignment->{"Columns" -> {{Left}}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}], Alignment->Left, BaselinePosition->Baseline, FrameMargins->{{5, 4}, {0, 4}}], "DialogStyle", FontFamily->"Roboto", FontSize->11]} }, BaselinePosition->{1, 1}, GridBoxAlignment->{"Columns" -> {{Left}}, "Rows" -> {{Baseline}}}, GridBoxDividers->{"Columns" -> {{None}}, "Rows" -> {False, { GrayLevel[0.8]}, False}}, GridBoxItemSize->{"Columns" -> {{Automatic}}, "Rows" -> {{Automatic}}}]}, Dynamic[Typeset`open], BaselinePosition->Baseline, ImageSize->Automatic], Background->RGBColor[ 0.9686274509803922, 0.9764705882352941, 0.984313725490196], BaselinePosition->Baseline, DefaultBaseStyle->{}, FrameMargins->{{0, 0}, {1, 0}}, FrameStyle->RGBColor[ 0.8313725490196079, 0.8470588235294118, 0.8509803921568627], RoundingRadius->4]], {"FunctionResourceBox", RGBColor[0.8745098039215686, 0.2784313725490196, 0.03137254901960784], "JordanTotient"}, TagBoxNote->"FunctionResourceBox"], ResourceFunction["JordanTotient"], BoxID -> "JordanTotient", Selectable->False][j, n]\)\)]
Out[6]=

Properties and Relations (5) 

JordanTotient[1,n] is the same as EulerPhi[n]:

In[7]:=
ResourceFunction["JordanTotient"][1, Range[20]] == EulerPhi[Range[20]]
Out[7]=

JordanTotient is a multiplicative function:

In[8]:=
With[{k = 5, m = 10, n = 13}, ResourceFunction["JordanTotient"][k, m] ResourceFunction[ "JordanTotient"][k, n] == ResourceFunction["JordanTotient"][k, m n] && CoprimeQ[m, n]]
Out[8]=

where p is prime:

In[9]:=
With[{k = 3, n = 5}, ResourceFunction["JordanTotient"][k, n] == n^k Product[1 - p^-k, {p, Select[Divisors[n], PrimeQ]}]]
Out[9]=

JordanTotient[k,n] counts the number of k-tuples ≤n that form a coprime (k+1)-tuple together with n:

In[10]:=
With[{k = 3, n = 5}, Length[Select[Tuples[Range[n], {k}], Apply[GCD, Append[#, n]] == 1 &]]]
Out[10]=
In[11]:=
ResourceFunction["JordanTotient"][3, 5]
Out[11]=

The power function can be expressed as a divisor sum of Jordan totients:

In[12]:=
With[{k = 3, n = 5}, DivisorSum[n, ResourceFunction["JordanTotient"][k, #] &] == n^k]
Out[12]=

Neat Examples (1) 

Plot the Ulam spiral with numbers colored based on the values of JordanTotient:

In[13]:=
ulam[n_] := Partition[Permute[Range[n^2], Accumulate[Take[Flatten[{{n^2 + 1}/2, Table[(-1)^j i, {j, n}, {i, {-1, n}}, {j}]}], n^2]]], n]
In[14]:=
Manipulate[ ArrayPlot[ResourceFunction["JordanTotient"][k, ulam[141]], ColorFunction -> Hue], {k, 1, 10, 1}, ContinuousAction -> False, SaveDefinitions -> True]
Out[14]=

Version History

  • 1.0.0 – 10 March 2021

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