(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 11.1' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       158,          7]
NotebookDataLength[     11568,        342]
NotebookOptionsPosition[      8878,        268]
NotebookOutlinePosition[      9253,        285]
CellTagsIndexPosition[      9210,        282]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{

Cell[CellGroupData[{
Cell["Moebius Transform", "Title",ExpressionUUID->"81117ff4-cd9b-4fbb-ad3a-51455b648cfd"],

Cell[CellGroupData[{

Cell["Author", "Subsection",ExpressionUUID->"78d153d8-da32-462d-b502-b120009e9cd7"],

Cell["\<\
Eric W. Weisstein
May 8, 2008\
\>", "Text",ExpressionUUID->"19c323ef-ca42-49fe-aea7-11e1c875d25c"],

Cell[TextData[{
 "This notebook downloaded from ",
 ButtonBox["http://mathworld.wolfram.com/notebooks/Combinatorics/\
MoebiusTransform.nb",
  BaseStyle->"Hyperlink",
  ButtonData:>{
    URL["http://mathworld.wolfram.com/notebooks/Combinatorics/\
MoebiusTransform.nb"], None}],
 "."
}], "Text",ExpressionUUID->"b1f8df44-ff55-4eb1-bdd0-77993857f388"],

Cell[TextData[{
 "For more information, see Eric's ",
 StyleBox["MathWorld",
  FontSlant->"Italic"],
 " entry ",
 ButtonBox["http://mathworld.wolfram.com/MoebiusTransform.html",
  BaseStyle->"Hyperlink",
  ButtonData:>{
    URL["http://mathworld.wolfram.com/MoebiusTransform.html"], None}],
 "."
}], "Text",ExpressionUUID->"a96a9c1c-3bcb-4882-96cc-e4ee6b01afd8"],

Cell["\<\
\[Copyright]2008 Wolfram Research, Inc. except for portions noted otherwise\
\>", "Text",ExpressionUUID->"034b19e0-81e9-47ff-ab76-f7e4b4053e46"]
}, Open  ]],

Cell[CellGroupData[{

Cell["Examples", "Section",ExpressionUUID->"9647d2f5-dae5-4f11-a60f-4a6a731fc84b"],

Cell["Number of divisors", "Text",ExpressionUUID->"2a717cc7-266c-4882-8bbe-e1c0621f7b70"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"MoebiusTransform", "[", 
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{"DivisorSigma", "[", 
     RowBox[{"0", ",", "n"}], "]"}], ",", 
    RowBox[{"{", 
     RowBox[{"n", ",", "10"}], "}"}]}], "]"}], "]"}]], "Input",ExpressionUUID-\
>"89e33c94-15ea-4316-932d-94f89ad98219"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "1", ",", "1", ",", "1", ",", "1", ",", "1", ",", "1", ",", "1", ",", "1", 
   ",", "1", ",", "1"}], "}"}]], "Output",ExpressionUUID->"2fbc665b-f39a-4fcd-\
b621-8ed5badb28fd"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"{", 
   RowBox[{
    RowBox[{"InverseMoebiusTransform", "[", 
     RowBox[{"Table", "[", 
      RowBox[{"1", ",", 
       RowBox[{"{", "20", "}"}]}], "]"}], "]"}], ",", "\[IndentingNewLine]", 
    RowBox[{"Table", "[", 
     RowBox[{
      RowBox[{"DivisorSigma", "[", 
       RowBox[{"0", ",", "n"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"n", ",", "20"}], "}"}]}], "]"}]}], "}"}], "//", 
  "ColumnForm"}]], "Input",ExpressionUUID->"3a293716-6756-448c-9a86-\
9799d4f607dd"],

Cell[BoxData[
 InterpretationBox[GridBox[{
    {
     RowBox[{"{", 
      RowBox[{
      "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "4", ",", "2", ",", 
       "4", ",", "3", ",", "4", ",", "2", ",", "6", ",", "2", ",", "4", ",", 
       "4", ",", "5", ",", "2", ",", "6", ",", "2", ",", "6"}], "}"}]},
    {
     RowBox[{"{", 
      RowBox[{
      "1", ",", "2", ",", "2", ",", "3", ",", "2", ",", "4", ",", "2", ",", 
       "4", ",", "3", ",", "4", ",", "2", ",", "6", ",", "2", ",", "4", ",", 
       "4", ",", "5", ",", "2", ",", "6", ",", "2", ",", "6"}], "}"}]}
   },
   BaselinePosition->{Baseline, {1, 1}},
   GridBoxAlignment->{
    "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, 
     "RowsIndexed" -> {}}],
  ColumnForm[{{1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6}, {
   1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6}}],
  Editable->False]], "Output",ExpressionUUID->"cb66592e-ec19-4e04-8067-\
00d1e367555c"]
}, Open  ]],

Cell["Totient function", "Text",ExpressionUUID->"7b5acfe4-6642-4872-9a06-1d925d392481"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"{", 
   RowBox[{
    RowBox[{"MoebiusTransform", "[", 
     RowBox[{"Range", "[", "20", "]"}], "]"}], ",", "\[IndentingNewLine]", 
    RowBox[{"Table", "[", 
     RowBox[{
      RowBox[{"EulerPhi", "[", "n", "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"n", ",", "20"}], "}"}]}], "]"}]}], "}"}], "//", 
  "ColumnForm"}]], "Input",ExpressionUUID->"39b885d1-1500-4d93-bd2e-\
733af587f0a7"],

Cell[BoxData[
 InterpretationBox[GridBox[{
    {
     RowBox[{"{", 
      RowBox[{
      "1", ",", "1", ",", "2", ",", "2", ",", "4", ",", "2", ",", "6", ",", 
       "4", ",", "6", ",", "4", ",", "10", ",", "4", ",", "12", ",", "6", ",",
        "8", ",", "8", ",", "16", ",", "6", ",", "18", ",", "8"}], "}"}]},
    {
     RowBox[{"{", 
      RowBox[{
      "1", ",", "1", ",", "2", ",", "2", ",", "4", ",", "2", ",", "6", ",", 
       "4", ",", "6", ",", "4", ",", "10", ",", "4", ",", "12", ",", "6", ",",
        "8", ",", "8", ",", "16", ",", "6", ",", "18", ",", "8"}], "}"}]}
   },
   BaselinePosition->{Baseline, {1, 1}},
   GridBoxAlignment->{
    "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, 
     "RowsIndexed" -> {}}],
  ColumnForm[{{1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 
   8}, {1, 1, 2, 2, 4, 2, 6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16, 6, 18, 8}}],
  Editable->False]], "Output",ExpressionUUID->"093d5976-469c-443c-bcf0-\
6f1a36c38b71"]
}, Open  ]],

Cell["Sum of two squares", "Text",ExpressionUUID->"e5046ed3-6148-4726-a599-61664c3915c1"],

Cell[BoxData[{
 RowBox[{
  RowBox[{"b", "[", 
   RowBox[{"n_Integer", "?", "EvenQ"}], "]"}], ":=", "0"}], "\n", 
 RowBox[{
  RowBox[{"b", "[", 
   RowBox[{"n_Integer", "?", "OddQ"}], "]"}], ":=", 
  RowBox[{"4", 
   RowBox[{
    RowBox[{"(", 
     RowBox[{"-", "1"}], ")"}], "^", 
    RowBox[{"(", 
     RowBox[{
      RowBox[{"(", 
       RowBox[{"n", "-", "1"}], ")"}], "/", "2"}], ")"}]}]}]}]}], "Input",Expr\
essionUUID->"b10a880e-b6de-466b-857b-b34837e9d976"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{
  RowBox[{"{", 
   RowBox[{
    RowBox[{"InverseMoebiusTransform", "[", 
     RowBox[{"Table", "[", 
      RowBox[{
       RowBox[{"b", "[", "n", "]"}], ",", 
       RowBox[{"{", 
        RowBox[{"n", ",", "20"}], "}"}]}], "]"}], "]"}], ",", 
    "\[IndentingNewLine]", 
    RowBox[{"Table", "[", 
     RowBox[{
      RowBox[{"SquaresR", "[", 
       RowBox[{"2", ",", "n"}], "]"}], ",", 
      RowBox[{"{", 
       RowBox[{"n", ",", "20"}], "}"}]}], "]"}]}], "}"}], "//", 
  "ColumnForm"}]], "Input",ExpressionUUID->"da52a9ba-d4b5-47f7-9601-\
536add4510eb"],

Cell[BoxData[
 InterpretationBox[GridBox[{
    {
     RowBox[{"{", 
      RowBox[{
      "4", ",", "4", ",", "0", ",", "4", ",", "8", ",", "0", ",", "0", ",", 
       "4", ",", "4", ",", "8", ",", "0", ",", "0", ",", "8", ",", "0", ",", 
       "0", ",", "4", ",", "8", ",", "4", ",", "0", ",", "8"}], "}"}]},
    {
     RowBox[{"{", 
      RowBox[{
      "4", ",", "4", ",", "0", ",", "4", ",", "8", ",", "0", ",", "0", ",", 
       "4", ",", "4", ",", "8", ",", "0", ",", "0", ",", "8", ",", "0", ",", 
       "0", ",", "4", ",", "8", ",", "4", ",", "0", ",", "8"}], "}"}]}
   },
   BaselinePosition->{Baseline, {1, 1}},
   GridBoxAlignment->{
    "Columns" -> {{Left}}, "ColumnsIndexed" -> {}, "Rows" -> {{Baseline}}, 
     "RowsIndexed" -> {}}],
  ColumnForm[{{4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8}, {
   4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8}}],
  Editable->False]], "Output",ExpressionUUID->"20cb9eff-d60b-498e-90dd-\
396a61e9b8bf"]
}, Open  ]],

Cell["Number of distinct primes dividing n (M0056)", "Text",ExpressionUUID->"40640417-c08f-4dc6-876e-2ec531fe754a"],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"InverseMoebiusTransform", "[", 
  RowBox[{"Table", "[", 
   RowBox[{
    RowBox[{"p", "[", "n", "]"}], ",", 
    RowBox[{"{", 
     RowBox[{"n", ",", "50"}], "}"}]}], "]"}], "]"}]], "Input",ExpressionUUID-\
>"ae2fb66c-4add-4e80-a8fd-22378618ed9b"],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{
  "0", ",", "1", ",", "1", ",", "1", ",", "1", ",", "2", ",", "1", ",", "1", 
   ",", "1", ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "2", ",", 
   "1", ",", "1", ",", "2", ",", "1", ",", "2", ",", "2", ",", "2", ",", "1", 
   ",", "2", ",", "1", ",", "2", ",", "1", ",", "2", ",", "1", ",", "3", ",", 
   "1", ",", "1", ",", "2", ",", "2", ",", "2", ",", "2", ",", "1", ",", "2", 
   ",", "2", ",", "2", ",", "1", ",", "3", ",", "1", ",", "2", ",", "2", ",", 
   "2", ",", "1", ",", "2", ",", "1", ",", "2"}], "}"}]], "Output",ExpressionU\
UID->"3f843c96-83d0-433e-a852-612582a6db88"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
},
WindowSize->{606, 858},
WindowMargins->{{Automatic, 193}, {Automatic, 21}},
ShowSelection->True,
FrontEndVersion->"11.1 for Mac OS X x86 (32-bit, 64-bit Kernel) (June 2, \
2017)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[CellGroupData[{
Cell[580, 22, 89, 0, 92, "Title", "ExpressionUUID" -> \
"81117ff4-cd9b-4fbb-ad3a-51455b648cfd"],
Cell[CellGroupData[{
Cell[694, 26, 83, 0, 44, "Subsection", "ExpressionUUID" -> \
"78d153d8-da32-462d-b502-b120009e9cd7"],
Cell[780, 28, 108, 3, 49, "Text", "ExpressionUUID" -> \
"19c323ef-ca42-49fe-aea7-11e1c875d25c"],
Cell[891, 33, 348, 9, 49, "Text", "ExpressionUUID" -> \
"b1f8df44-ff55-4eb1-bdd0-77993857f388"],
Cell[1242, 44, 362, 10, 49, "Text", "ExpressionUUID" -> \
"a96a9c1c-3bcb-4882-96cc-e4ee6b01afd8"],
Cell[1607, 56, 154, 2, 30, "Text", "ExpressionUUID" -> \
"034b19e0-81e9-47ff-ab76-f7e4b4053e46"]
}, Open  ]],
Cell[CellGroupData[{
Cell[1798, 63, 82, 0, 64, "Section", "ExpressionUUID" -> \
"9647d2f5-dae5-4f11-a60f-4a6a731fc84b"],
Cell[1883, 65, 89, 0, 30, "Text", "ExpressionUUID" -> \
"2a717cc7-266c-4882-8bbe-e1c0621f7b70"],
Cell[CellGroupData[{
Cell[1997, 69, 301, 8, 32, "Input", "ExpressionUUID" -> \
"89e33c94-15ea-4316-932d-94f89ad98219"],
Cell[2301, 79, 217, 5, 32, "Output", "ExpressionUUID" -> \
"2fbc665b-f39a-4fcd-b621-8ed5badb28fd"]
}, Open  ]],
Cell[CellGroupData[{
Cell[2555, 89, 519, 15, 54, "Input", "ExpressionUUID" -> \
"3a293716-6756-448c-9a86-9799d4f607dd"],
Cell[3077, 106, 981, 22, 51, "Output", "ExpressionUUID" -> \
"cb66592e-ec19-4e04-8067-00d1e367555c"]
}, Open  ]],
Cell[4073, 131, 87, 0, 30, "Text", "ExpressionUUID" -> \
"7b5acfe4-6642-4872-9a06-1d925d392481"],
Cell[CellGroupData[{
Cell[4185, 135, 425, 12, 54, "Input", "ExpressionUUID" -> \
"39b885d1-1500-4d93-bd2e-733af587f0a7"],
Cell[4613, 149, 997, 22, 51, "Output", "ExpressionUUID" -> \
"093d5976-469c-443c-bcf0-6f1a36c38b71"]
}, Open  ]],
Cell[5625, 174, 89, 0, 30, "Text", "ExpressionUUID" -> \
"e5046ed3-6148-4726-a599-61664c3915c1"],
Cell[5717, 176, 464, 15, 54, "Input", "ExpressionUUID" -> \
"b10a880e-b6de-466b-857b-b34837e9d976"],
Cell[CellGroupData[{
Cell[6206, 195, 582, 18, 54, "Input", "ExpressionUUID" -> \
"da52a9ba-d4b5-47f7-9601-536add4510eb"],
Cell[6791, 215, 981, 22, 51, "Output", "ExpressionUUID" -> \
"20cb9eff-d60b-498e-90dd-396a61e9b8bf"]
}, Open  ]],
Cell[7787, 240, 115, 0, 30, "Text", "ExpressionUUID" -> \
"40640417-c08f-4dc6-876e-2ec531fe754a"],
Cell[CellGroupData[{
Cell[7927, 244, 271, 7, 32, "Input", "ExpressionUUID" -> \
"ae2fb66c-4add-4e80-a8fd-22378618ed9b"],
Cell[8201, 253, 637, 10, 70, "Output", "ExpressionUUID" -> \
"3f843c96-83d0-433e-a852-612582a6db88"]
}, Open  ]]
}, Open  ]]
}, Open  ]]
}
]
*)

(* End of internal cache information *)