(* Content-type: application/mathematica *)

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

(* CreatedBy='Mathematica 7.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     18460,        538]
NotebookOptionsPosition[     17358,        497]
NotebookOutlinePosition[     17725,        513]
CellTagsIndexPosition[     17682,        510]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
 StyleBox[
  RowBox[{"How", " ", "many", " ", "distinct", " ", "prime", " ", 
   RowBox[{"factors", "?"}]}], "Subtitle"]], "Input",
 CellChangeTimes->{{3.463699480484375*^9, 3.463699490484375*^9}, {
  3.4637560361230717`*^9, 3.4637560425015697`*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
  "number", " ", "=", " ", "5487525252462375652364513298043621345687"}], 
  ";"}]], "Input",
 CellChangeTimes->{{3.463698653921875*^9, 3.463698659828125*^9}, {
   3.463699072875*^9, 3.463699079640625*^9}, {3.46369916290625*^9, 
   3.4636992610625*^9}, {3.46369936909375*^9, 3.463699369296875*^9}, {
   3.463699424734375*^9, 3.46369943346875*^9}, {3.463756083652256*^9, 
   3.4637560838543916`*^9}, {3.46382873565625*^9, 3.463828738796875*^9}, {
   3.46382876971875*^9, 3.463828769984375*^9}, 3.463829167515625*^9}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"findingprimefactors", "[", 
    RowBox[{"numb_", ",", "option_"}], "]"}], " ", ":=", " ", 
   RowBox[{"Module", "[", 
    RowBox[{
     RowBox[{"{", "}"}], ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"temp", " ", "=", " ", 
       RowBox[{"FactorInteger", "[", "numb", "]"}]}], ";", 
      "\[IndentingNewLine]", 
      RowBox[{"numfactors", " ", "=", " ", 
       RowBox[{"Length", "[", "temp", "]"}]}], ";", "\[IndentingNewLine]", 
      "\[IndentingNewLine]", 
      RowBox[{"If", "[", 
       RowBox[{
        RowBox[{"option", " ", "\[Equal]", " ", "1"}], ",", 
        "\[IndentingNewLine]", 
        RowBox[{"{", "\[IndentingNewLine]", 
         RowBox[{
          RowBox[{"Print", "[", 
           RowBox[{
           "\"\<Number is \>\"", ",", " ", "numb", ",", " ", "\"\<.\>\""}], 
           "]"}], ";", "\[IndentingNewLine]", 
          RowBox[{"Print", "[", 
           RowBox[{
           "\"\<Number of prime factors is \>\"", ",", " ", "numfactors", ",",
             " ", "\"\<.\>\""}], "]"}], ";", "\[IndentingNewLine]", 
          RowBox[{"Print", "[", 
           RowBox[{"\"\<Expected number of prime factors is \>\"", ",", "  ", 
            RowBox[{"1.0", " ", 
             RowBox[{"Log", "[", " ", 
              RowBox[{"Log", "[", "numb", "]"}], "]"}]}], ",", " ", 
            "\"\<.\>\""}], "]"}], ";", "\[IndentingNewLine]", 
          RowBox[{"Print", "[", 
           RowBox[{
           "\"\<Prime factorization is \>\"", ",", "temp", ",", " ", 
            "\"\<.\>\""}], "]"}], ";"}], "\[IndentingNewLine]", "}"}], ",", 
        " ", 
        RowBox[{"Return", "[", "numfactors", "]"}]}], "]"}], ";"}]}], 
    "\[IndentingNewLine]", "]"}]}], ";"}]], "Input",
 CellChangeTimes->{{3.4636986674375*^9, 3.463698833484375*^9}, 
   3.463698904078125*^9, {3.463698940796875*^9, 3.463699043484375*^9}, {
   3.463699129125*^9, 3.463699152296875*^9}, {3.463699265109375*^9, 
   3.4636992888125*^9}, {3.46369935540625*^9, 3.463699364140625*^9}, {
   3.4636994140625*^9, 3.463699416125*^9}, {3.46370101915625*^9, 
   3.46370106009375*^9}, {3.46370120903125*^9, 3.463701211921875*^9}, {
   3.4637559015369587`*^9, 3.463756026362954*^9}, {3.4637560685858374`*^9, 
   3.463756069614092*^9}, {3.4637562422954226`*^9, 3.46375626187913*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"findingprimefactors", "[", 
  RowBox[{"number", " ", ",", "1"}], "]"}]], "Input",
 CellChangeTimes->{{3.463698977828125*^9, 3.46369899046875*^9}, {
  3.46369907525*^9, 3.463699075796875*^9}, {3.463700744625*^9, 
  3.463700748203125*^9}, {3.46370097209375*^9, 3.463700986375*^9}, {
  3.46370106609375*^9, 3.46370106696875*^9}, {3.463756046058959*^9, 
  3.4637560579790945`*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Number is \"\>", "\[InvisibleSpace]", 
   "5487525252462375652364513298043621345687", 
   "\[InvisibleSpace]", "\<\".\"\>"}],
  SequenceForm["Number is ", 5487525252462375652364513298043621345687, "."],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.463828778515625*^9, 3.46382916984375*^9, {3.464037016453125*^9, 
   3.464037019859375*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Number of prime factors is \"\>", "\[InvisibleSpace]", "5", 
   "\[InvisibleSpace]", "\<\".\"\>"}],
  SequenceForm["Number of prime factors is ", 5, "."],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.463828778515625*^9, 3.46382916984375*^9, {3.464037016453125*^9, 
   3.464037019875*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Expected number of prime factors is \"\>", "\[InvisibleSpace]",
    "4.516374993566843`", "\[InvisibleSpace]", "\<\".\"\>"}],
  SequenceForm["Expected number of prime factors is ", 4.516374993566843, "."],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.463828778515625*^9, 3.46382916984375*^9, {3.464037016453125*^9, 
   3.464037019875*^9}}],

Cell[BoxData[
 InterpretationBox[
  RowBox[{"\<\"Prime factorization is \"\>", "\[InvisibleSpace]", 
   RowBox[{"{", 
    RowBox[{
     RowBox[{"{", 
      RowBox[{"7", ",", "1"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"11", ",", "2"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"6619", ",", "1"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"101399", ",", "1"}], "}"}], ",", 
     RowBox[{"{", 
      RowBox[{"9653105849003586101891476541", ",", "1"}], "}"}]}], "}"}], 
   "\[InvisibleSpace]", "\<\".\"\>"}],
  SequenceForm[
  "Prime factorization is ", {{7, 1}, {11, 2}, {6619, 1}, {101399, 1}, {
    9653105849003586101891476541, 1}}, "."],
  Editable->False]], "Print",
 CellChangeTimes->{
  3.463828778515625*^9, 3.46382916984375*^9, {3.464037016453125*^9, 
   3.464037019875*^9}}]
}, Open  ]]
}, Open  ]],

Cell[BoxData[""], "Input",
 CellChangeTimes->{{3.463700945859375*^9, 3.463700946265625*^9}}],

Cell[BoxData[
 RowBox[{
  RowBox[{
   RowBox[{"histogramfactors", "[", 
    RowBox[{"start_", ",", "numdo_"}], "]"}], " ", ":=", " ", 
   RowBox[{"Module", "[", 
    RowBox[{
     RowBox[{"{", "}"}], ",", "\[IndentingNewLine]", 
     RowBox[{
      RowBox[{"output", " ", "=", " ", 
       RowBox[{"{", "}"}]}], ";", "\[IndentingNewLine]", 
      RowBox[{"For", "[", 
       RowBox[{
        RowBox[{"n", " ", "=", " ", "1"}], ",", " ", 
        RowBox[{"n", " ", "\[LessEqual]", " ", "numdo"}], ",", " ", 
        RowBox[{"n", "++"}], ",", " ", 
        RowBox[{"output", " ", "=", " ", 
         RowBox[{"AppendTo", "[", 
          RowBox[{"output", ",", " ", 
           RowBox[{"findingprimefactors", "[", " ", 
            RowBox[{
             RowBox[{"start", " ", "+", " ", "n"}], ",", "0"}], "]"}]}], 
          "]"}]}]}], "]"}], ";", "\[IndentingNewLine]", 
      RowBox[{"Print", "[", 
       RowBox[{"Histogram", "[", 
        RowBox[{"output", ",", "Automatic", ",", " ", "\"\<Probability\>\""}],
         "]"}], "]"}], ";"}]}], "\[IndentingNewLine]", "]"}]}], 
  ";"}]], "Input",
 CellChangeTimes->{{3.46370094878125*^9, 3.46370095934375*^9}, {
  3.463700990890625*^9, 3.46370100815625*^9}, {3.463701076046875*^9, 
  3.463701097515625*^9}, {3.46370122646875*^9, 3.4637012760625*^9}, {
  3.46370139040625*^9, 3.46370141709375*^9}, {3.463701501109375*^9, 
  3.4637015055625*^9}, {3.46375610081522*^9, 3.463756143872034*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Timing", "[", 
  RowBox[{"histogramfactors", "[", 
   RowBox[{
   "5487525252462375634352364513298043621345687989991218989811", ",", 
    "1000"}], "]"}], "]"}]], "Input",
 CellChangeTimes->{{3.4637007899375*^9, 3.463700811578125*^9}, {
  3.46370089390625*^9, 3.4637008973125*^9}, {3.463700928890625*^9, 
  3.4637009304375*^9}, {3.463701280078125*^9, 3.463701305578125*^9}, {
  3.463701356*^9, 3.463701361828125*^9}, {3.463701426515625*^9, 
  3.463701462109375*^9}, {3.463701547046875*^9, 3.463701547203125*^9}, {
  3.4637561543440204`*^9, 3.4637561880313177`*^9}, {3.4637563010045557`*^9, 
  3.463756324714952*^9}, {3.46382879690625*^9, 3.463828798203125*^9}, {
  3.463829186265625*^9, 3.463829192375*^9}, {3.464037100796875*^9, 
  3.464037170703125*^9}}],

Cell[BoxData[
 GraphicsBox[
  {RGBColor[0.798413061722744, 0.824719615472648, 0.968322270542458], 
   EdgeForm[Opacity[0.7]], {}, 
   {RGBColor[0.798413061722744, 0.824719615472648, 0.968322270542458], 
    EdgeForm[Opacity[0.7]], {}, 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{1, 0}, {2, 0.008}]},
        ImageSizeCache->{{48.33367346938776, 76.41734693877551}, {
         90.55016938191983, 97.68819151024776}}],
       StatusArea[#, 0.008]& ,
       TagBoxNote->"0.008"],
      StyleBox["0.008`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.008, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{2, 0}, {3, 0.045}]},
        ImageSizeCache->{{75.91734693877551, 104.00102040816327`}, {
         59.849317038403164`, 97.68819151024776}}],
       StatusArea[#, 0.045]& ,
       TagBoxNote->"0.045"],
      StyleBox["0.045`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.045, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{3, 0}, {4, 0.133}]},
        ImageSizeCache->{{103.50102040816327`, 
         131.58469387755102`}, {-13.168926373204059`, 97.68819151024776}}],
       StatusArea[#, 0.133]& ,
       TagBoxNote->"0.133"],
      StyleBox["0.133`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.133, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{4, 0}, {5, 0.171}]},
        ImageSizeCache->{{131.08469387755102`, 
         159.16836734693877`}, {-44.699531482761714`, 97.68819151024776}}],
       StatusArea[#, 0.171]& ,
       TagBoxNote->"0.171"],
      StyleBox["0.171`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.171, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{5, 0}, {6, 0.226}]},
        ImageSizeCache->{{158.66836734693877`, 
         186.75204081632654`}, {-90.33593361501623, 97.68819151024776}}],
       StatusArea[#, 0.226]& ,
       TagBoxNote->"0.226"],
      StyleBox["0.226`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.226, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{6, 0}, {7, 0.195}]},
        ImageSizeCache->{{186.25204081632654`, 
         214.3357142857143}, {-64.6135978677455, 97.68819151024776}}],
       StatusArea[#, 0.195]& ,
       TagBoxNote->"0.195"],
      StyleBox["0.195`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.195, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{7, 0}, {8, 0.139}]},
        ImageSizeCache->{{213.8357142857143, 
         241.91938775510204`}, {-18.14744296945001, 97.68819151024776}}],
       StatusArea[#, 0.139]& ,
       TagBoxNote->"0.139"],
      StyleBox["0.139`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.139, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{8, 0}, {9, 0.044}]},
        ImageSizeCache->{{241.41938775510204`, 269.5030612244898}, {
         60.679069804444154`, 97.68819151024776}}],
       StatusArea[#, 0.044]& ,
       TagBoxNote->"0.044"],
      StyleBox["0.044`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.044, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{9, 0}, {10, 0.024}]},
        ImageSizeCache->{{269.0030612244898, 297.08673469387753`}, {
         77.27412512526398, 97.68819151024776}}],
       StatusArea[#, 0.024]& ,
       TagBoxNote->"0.024"],
      StyleBox["0.024`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.024, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{10, 0}, {11, 0.013}]},
        ImageSizeCache->{{296.58673469387753`, 324.6704081632653}, {
         86.40140555171487, 97.68819151024776}}],
       StatusArea[#, 0.013]& ,
       TagBoxNote->"0.013"],
      StyleBox["0.013`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.013, {
        GrayLevel[0]}], "Tooltip"]& ], 
    TagBox[
     TooltipBox[
      TagBox[
       DynamicBox[{
         FEPrivate`If[
          CurrentValue["MouseOver"], 
          EdgeForm[{
            GrayLevel[0.5], 
            AbsoluteThickness[1.5], 
            Opacity[0.66]}], {}, {}], 
         RectangleBox[{11, 0}, {12, 0.002}]},
        ImageSizeCache->{{324.1704081632653, 352.2540816326531}, {
         95.52868597816578, 97.68819151024776}}],
       StatusArea[#, 0.002]& ,
       TagBoxNote->"0.002"],
      StyleBox["0.002`", {
        GrayLevel[0]}, StripOnInput -> False]],
     Annotation[#, 
      Style[0.002, {
        GrayLevel[0]}], "Tooltip"]& ]}, {}, {}},
  AspectRatio->NCache[GoldenRatio^(-1), 0.6180339887498948],
  Axes->{True, True},
  AxesOrigin->{0, 0},
  FrameTicks->{{Automatic, Automatic}, {Automatic, Automatic}},
  PlotRange->{{0, 12}, {All, All}},
  PlotRangePadding->{{
     Scaled[0.02], 
     Scaled[0.02]}, {
     Scaled[0.02], 
     Scaled[0.1]}},
  Ticks->{Automatic, Automatic}]], "Print",
 CellChangeTimes->{{3.463829189921875*^9, 3.46382919878125*^9}, 
   3.4640370423125*^9, {3.464037109953125*^9, 3.46403716603125*^9}, 
   3.464041853484375*^9}],

Cell[BoxData[
 RowBox[{"{", 
  RowBox[{"4273.36`", ",", "Null"}], "}"}]], "Output",
 CellChangeTimes->{{3.46403711815625*^9, 3.46403716603125*^9}, 
   3.464041853484375*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"4273", "/", "60."}]], "Input",
 CellChangeTimes->{{3.46404239834375*^9, 3.46404240409375*^9}}],

Cell[BoxData["71.21666666666667`"], "Output",
 CellChangeTimes->{3.464042404609375*^9}]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"Log", "[", 
  "5487525252462375634352364513298043621345687989991218989811.", 
  "]"}]], "Input",
 CellChangeTimes->{{3.464044090171875*^9, 3.464044107625*^9}}],

Cell[BoxData["132.\
9498276808667724924228964505685158271377543592207661199914858554588`59.\
86306431055817"], "Output",
 CellChangeTimes->{{3.464044096828125*^9, 3.464044108125*^9}}]
}, Open  ]],

Cell[BoxData[{
 RowBox[{"n", " ", "=", " ", "10"}], "\[IndentingNewLine]", 
 RowBox[{"m", " ", "=", " ", "10"}], "\[IndentingNewLine]", 
 RowBox[{"p", " ", "=", " ", 
  RowBox[{"1", "/", "2"}]}], "\[IndentingNewLine]", 
 RowBox[{"n", " ", 
  RowBox[{"(", 
   RowBox[{"1", "/", "2"}], ")"}]}]}], "Input",
 CellChangeTimes->{{3.46557861871875*^9, 3.46557863053125*^9}}]
},
WindowSize->{616, 861},
WindowMargins->{{103, Automatic}, {Automatic, 18}},
ShowSelection->True,
FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 264, 5, 44, "Input"],
Cell[812, 27, 549, 10, 31, "Input"],
Cell[1364, 39, 2318, 49, 272, "Input"],
Cell[CellGroupData[{
Cell[3707, 92, 401, 7, 31, "Input"],
Cell[CellGroupData[{
Cell[4133, 103, 397, 9, 23, "Print"],
Cell[4533, 114, 346, 8, 23, "Print"],
Cell[4882, 124, 397, 8, 23, "Print"],
Cell[5282, 134, 801, 22, 43, "Print"]
}, Open  ]]
}, Open  ]],
Cell[6110, 160, 92, 1, 31, "Input"],
Cell[6205, 163, 1436, 32, 132, "Input"],
Cell[CellGroupData[{
Cell[7666, 199, 780, 14, 72, "Input"],
Cell[8449, 215, 7696, 239, 237, "Print"],
Cell[16148, 456, 173, 4, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[16358, 465, 118, 2, 31, "Input"],
Cell[16479, 469, 87, 1, 30, "Output"]
}, Open  ]],
Cell[CellGroupData[{
Cell[16603, 475, 183, 4, 31, "Input"],
Cell[16789, 481, 183, 3, 30, "Output"]
}, Open  ]],
Cell[16987, 487, 367, 8, 92, "Input"]
}
]
*)

(* End of internal cache information *)