(*
(c) 2018 Alex Kontorovich

This file computes Omega for Lucas and Fibonacci sequences and compares the two

Raw data for the Lucas sequence is taken from : 
http://mersennus.net/fibonacci/lucas.txt

The interpretation of the raw data is here:
http://mersennus.net/fibonacci/format.html

This extends OEIS A086599 (Number of prime factors in Lucas(n),counting multiplicity)

Raw data for Fibonacci sequence is taken from :
http://mersennus.net/fibonacci/fibonacci.txt

 The computation here is related to OEIS A134852

Many, many thanks to Danny Krashen for his very hard work in converting the raw data into the Omega data collected below
*)

(* First compute values of betas, b2 through b5 *)

k=2;
b2=x/.NSolve[x(1-Log[x]+Log[k])==k-1,x][[1]]

k=3;
b3=x/.NSolve[x(1-Log[x]+Log[k])==k-1,x][[1]]

k=4;
b4=x/.NSolve[x(1-Log[x]+Log[k])==k-1,x][[1]]


k=5;
b5=x/.NSolve[x(1-Log[x]+Log[k])==k-1,x][[1]]

(* This is the data (n, Omega(Lucas[n])) computed by Danny *)

lucData={{1,0},{2,1},{3,2},{4,1},{5,1},{6,3},{7,1},{8,1},{9,3},{10,2},{11,1},{12,3},{13,1},{14,2},{15,4},{16,1},{17,1},{18,5},{19,1},{20,2},{21,4},{22,3},{23,2},{24,3},{25,3},{26,2},{27,4},{28,3},{29,2},{30,5},{31,1},{32,2},{33,4},{34,3},{35,4},{36,4},{37,1},{38,2},{39,5},{40,3},{41,1},{42,6},{43,2},{44,4},{45,7},{46,3},{47,1},{48,4},{49,2},{50,4},{51,5},{52,3},{53,1},{54,7},{55,5},{56,2},{57,5},{58,3},{59,3},{60,6},{61,1},{62,2},{63,7},{64,2},{65,5},{66,6},{67,2},{68,2},{69,6},{70,4},{71,1},{72,4},{73,2},{74,3},{75,8},{76,2},{77,4},{78,5},{79,1},{80,2},{81,6},{82,4},{83,2},{84,7},{85,4},{86,2},{87,6},{88,3},{89,2},{90,8},{91,3},{92,3},{93,5},{94,4},{95,5},{96,4},{97,2},{98,4},{99,8},{100,4},{101,3},{102,7},{103,3},{104,4},{105,10},{106,4},{107,2},{108,6},{109,3},{110,5},{111,6},{112,4},{113,1},{114,7},{115,7},{116,3},{117,7},{118,3},{119,5},{120,6},{121,2},{122,4},{123,5},{124,4},{125,6},{126,10},{127,5},{128,2},{129,6},{130,5},{131,3},{132,9},{133,4},{134,4},{135,12},{136,2},{137,2},{138,8},{139,3},{140,5},{141,5},{142,6},{143,5},{144,5},{145,5},{146,3},{147,8},{148,3},{149,3},{150,9},{151,3},{152,3},{153,8},{154,6},{155,5},{156,7},{157,2},{158,5},{159,5},{160,4},{161,6},{162,10},{163,3},{164,2},{165,11},{166,4},{167,2},{168,5},{169,3},{170,7},{171,10},{172,2},{173,4},{174,8},{175,10},{176,4},{177,7},{178,2},{179,3},{180,9},{181,3},{182,5},{183,5},{184,4},{185,4},{186,6},{187,4},{188,4},{189,12},{190,7},{191,3},{192,7},{193,3},{194,5},{195,14},{196,6},{197,2},{198,10},{199,3},{200,6},{201,6},{202,2},{203,5},{204,6},{205,6},{206,3},{207,8},{208,4},{209,5},{210,11},{211,2},{212,3},{213,6},{214,6},{215,8},{216,8},{217,5},{218,3},{219,7},{220,6},{221,4},{222,9},{223,2},{224,4},{225,14},{226,5},{227,4},{228,7},{229,2},{230,6},{231,11},{232,6},{233,3},{234,10},{235,8},{236,5},{237,6},{238,7},{239,3},{240,7},{241,3},{242,6},{243,8},{244,4},{245,10},{246,8},{247,5},{248,3},{249,7},{250,7},{251,4},{252,11},{253,5},{254,4},{255,13},{256,3},{257,3},{258,7},{259,4},{260,8},{261,8},{262,4},{263,4},{264,7},{265,8},{266,7},{267,6},{268,4},{269,4},{270,12},{271,4},{272,5},{273,10},{274,5},{275,10},{276,9},{277,5},{278,3},{279,9},{280,7},{281,3},{282,9},{283,3},{284,2},{285,15},{286,7},{287,6},{288,7},{289,3},{290,9},{291,7},{292,6},{293,2},{294,11},{295,9},{296,6},{297,12},{298,4},{299,8},{300,12},{301,6},{302,4},{303,9},{304,8},{305,6},{306,12},{307,2},{308,9},{309,7},{310,5},{311,2},{312,9},{313,1},{314,3},{315,18},{316,3},{317,3},{318,10},{319,5},{320,4},{321,6},{322,9},{323,4},{324,9},{325,11},{326,3},{327,9},{328,2},{329,5},{330,13},{331,4},{332,3},{333,10},{334,6},{335,5},{336,11},{337,3},{338,7},{339,5},{340,7},{341,6},{342,14},{343,5},{344,4},{345,16},{346,3},{347,3},{348,7},{349,3},{350,10},{351,10},{352,4},{353,1},{354,6},{355,4},{356,3},{357,11},{358,4},{359,4},{360,9},{361,3},{362,4},{363,9},{364,10},{365,8},{366,10},{367,5},{368,4},{369,10},{370,7},{371,5},{372,8},{373,2},{374,10},{375,15},{376,6},{377,5},{378,15},{379,3},{380,5},{381,10},{382,4},{383,4},{384,7},{385,15},{386,4},{387,9},{388,4},{389,2},{390,11},{391,6},{392,5},{393,7},{394,5},{395,11},{396,15},{397,2},{398,3},{399,13},{400,6},{401,3},{402,9},{403,4},{404,4},{405,20},{406,8},{407,6},{408,7},{409,4},{410,9},{411,7},{412,5},{413,7},{414,13},{415,8},{416,5},{417,7},{418,8},{419,5},{420,12},{421,3},{422,5},{423,7},{424,5},{425,11},{426,12},{427,6},{428,5},{429,13},{430,5},{431,4},{432,9},{433,4},{434,8},{435,17},{436,3},{437,5},{438,9},{439,4},{440,8},{441,14},{442,8},{443,3},{444,10},{445,8},{446,4},{447,8},{448,6},{449,4},{450,15},{451,4},{452,5},{453,8},{454,6},{455,14},{456,9},{457,4},{458,2},{459,10},{460,7},{461,3},{462,14},{463,5},{464,3},{465,18},{466,6},{467,3},{468,9},{469,7},{470,8},{471,7},{472,4},{473,7},{474,12},{475,11},{476,6},{477,9},{478,3},{479,5},{480,7},{481,5},{482,4},{483,14},{484,7},{485,7},{486,12},{487,2},{488,3},{489,8},{490,10},{491,3},{492,9},{493,6},{494,6},{495,21},{496,3},{497,7},{498,9},{499,3},{500,7},{501,9},{502,4},{503,1},{504,7},{505,10},{506,10},{507,9},{508,3},{509,4},{510,14},{511,5},{512,4},{513,12},{514,3},{515,9},{516,8},{517,8},{518,6},{519,10},{520,9},{521,2},{522,15},{523,3},{524,3},{525,21},{526,4},{527,4},{528,9},{529,6},{530,6},{531,10},{532,8},{533,6},{534,8},{535,7},{536,6},{537,8},{538,3},{539,10},{540,14},{541,2},{542,4},{543,7},{544,5},{545,11},{546,13},{547,3},{548,6},{549,9},{550,10},{551,7},{552,11},{553,5},{554,6},{555,15},{556,5},{557,2},{558,14},{559,8},{560,9},{561,13},{562,5},{563,5},{564,9},{565,8},{566,3},{567,18},{568,5},{569,5},{570,16},{571,4},{572,9},{573,7},{574,9},{575,12},{576,11},{577,2},{578,7},{579,8},{580,7},{581,7},{582,10},{583,5},{584,4},{585,23},{586,2},{587,5},{588,12},{589,7},{590,6},{591,10},{592,3},{593,3},{594,16},{595,15},{596,4},{597,8},{598,6},{599,3},{600,12},{601,4},{602,8},{603,9},{604,7},{605,15},{606,5},{607,4},{608,5},{609,13},{610,8},{611,5},{612,9},{613,1},{614,3},{615,17},{616,7},{617,1},{618,11},{619,2},{620,8},{621,14},{622,6},{623,6},{624,10},{625,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(* This is the data (n, Omega(Fibonacci[n])) computed by Danny *)

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(* get sequence of just the Omegas *)

lucOmegas=Transpose[lucData][[2]];

(* For the Fibonaccis, we only get data for odd values and need to compute data for all values *)

fibOddOmegas=Transpose[fibOddData][[2]];

Length[lucOmegas]

Length[fibOddOmegas]

(* Computing Omega[Fibonacci[n]] from odd values *)
fibOmegas={};
For[n=1,n<10000,n++,
If[Mod[n,2]==1, (* if odd, then we already know the value *)
AppendTo[fibOmegas,fibOddOmegas[[(n+1)/2]]];
,
(* if even, then F_{2n}=F_n L_n, so the Omegas are added *)
AppendTo[fibOmegas,fibOmegas[[n/2]]+lucOmegas[[n/2]]];
];
];

Length[fibOmegas]

(* Test Fibonacci Omega values -- early values are off (Omega[1]=0, not 1) but doesn't matter; the rest is correct *)

Take[fibOmegas,30]
Table[PrimeOmega[Fibonacci[j]],{j,1,30}]

(* Test Lucas Omegas *)

Take[lucOmegas,30]
Table[PrimeOmega[LucasL[j]],{j,1,30}]

(* All is correct. *)

(* Now process data. First F_n L_n *)

ratioDataFL=Table[(fibOmegas[[j]]+lucOmegas[[j]])/Log[ Log[Fibonacci[j] LucasL[j]]],{j,2,9999}];
Show[
ListPlot[ratioDataFL,PlotRange->{All,{0,2}},PlotStyle->Directive[PointSize[0.01],Blue]]
,
Graphics[{Red,FontSize->42,Text[Subscript[β,2],{-300,0.3}]}],
Plot[b2,{x,0,10000},PlotStyle->Directive[Red,Thickness[0.005],Dotted]],AxesStyle->Thick,LabelStyle->Directive[Black,Bold,FontSize->36],PlotRange->{{-400,10000},{0,3}},ImageSize -> {958.85546875, Automatic}]

(* Next data: F_n F_n+1*)

ratioDataFF=Table[(fibOmegas[[j-1]]+fibOmegas[[j]])/Log[Log[Fibonacci[j-1]Fibonacci[j]]],{j,2,9999}];
Show[
ListPlot[ratioDataFF,PlotRange->{All,{0,2}},PlotStyle->Directive[Blue,PointSize[0.01]]]
,
Graphics[{Red,FontSize->42,Text[Subscript[β,3],{-300,b3-.05}]}],
Plot[b3,{x,0,10000},PlotStyle->Directive[Red,Thickness[0.005],Dotted]],AxesStyle->Thick,LabelStyle->Directive[Black,Bold,FontSize->36],PlotRange->{{-400,10000},{0,5}},ImageSize -> {958.85546875, Automatic}]

(* Next data: L_n L_n+1*)

ratioDataLL=Table[(lucOmegas[[j-1]]+lucOmegas[[j]])/Log[Log[LucasL[j-1]LucasL[j]]],{j,2,9999}];
Show[
ListPlot[ratioDataLL,PlotRange->{All,{0,2}},PlotStyle->Directive[Blue,PointSize[0.01]]]
,
Graphics[{Red,FontSize->42,Text[Subscript[β,2],{-300,b2-.05}]}],
Plot[b2,{x,0,10000},PlotStyle->Directive[Red,Thickness[0.005],Dotted]],AxesStyle->Thick,LabelStyle->Directive[Black,Bold,FontSize->36],PlotRange->{{-400,10000},{0,5}},ImageSize -> {958.85546875, Automatic}]

(* Next return to F_n L_n but separate according to even/odd n *)

ratioDataFLe=Table[{2j,(fibOmegas[[2j]]+lucOmegas[[2j]])/Log[ Log[Fibonacci[2j] LucasL[2j]]]},{j,2,4999}];
ratioDataFLo=Table[{2j-1,(fibOmegas[[2j-1]]+lucOmegas[[2j-1]])/Log[ Log[Fibonacci[2j-1] LucasL[2j-1]]]},{j,2,4999}];
Show[
ListPlot[ratioDataFLo,PlotRange->{All,{0,2}},PlotStyle->Directive[PointSize[Small],Blue]]
,
ListPlot[ratioDataFLe,PlotRange->{All,{0,2}},PlotStyle->Directive[PointSize[0.01],Blue]],
Graphics[{Red,FontSize->42,Text[Subscript[β,2],{-300,0.3}]}],
Graphics[{Red,FontSize->42,Text[Subscript[β,3],{-300,b3-.05}]}],
Plot[b2,{x,0,10000},PlotStyle->Directive[Red,Dotted]],
Plot[b3,{x,0,10000},PlotStyle->Directive[Red,Thickness[0.005],Dotted]],AxesStyle->Thick,LabelStyle->Directive[Black,Bold,FontSize->36],PlotRange->{{-400,10000},{0,3}},ImageSize -> {958.85546875, Automatic}]

(* Next F_2n F_{2n+2} = F_n L_n F_{n+1} L_{n+1} *)

ratioDataFLFL=Table[(lucOmegas[[j-1]]+lucOmegas[[j]]+fibOmegas[[j-1]]+fibOmegas[[j]])/Log[Log[Fibonacci[2j-2]Fibonacci[2j]]],{j,2,9999}];
Show[
ListPlot[ratioDataFLFL,PlotRange->{All,{0,2}},PlotStyle->Directive[Blue,PointSize[0.01]]]
,
Graphics[{Red,FontSize->42,Text[Subscript[β,4],{-300,b4-.05}]}],
Plot[b4,{x,0,10000},PlotStyle->Directive[Red,Thickness[0.005],Dotted]],AxesStyle->Thick,LabelStyle->Directive[Black,Bold,FontSize->36],PlotRange->{{-400,10000},{0,5}},ImageSize -> {958.85546875, Automatic}]

(* Next we import Mersenne factorizations from OEIS https://oeis.org/A046051/b046051.txt and factordb.com, thanks to Sean Irvine *)

TwoToNData=Take[Transpose[{{1,0},{2,1},{3,1},{4,2},{5,1},{6,3},{7,1},{8,3},{9,2},{10,3},{11,2},{12,5},{13,1},{14,3},{15,3},{16,4},{17,1},{18,6},{19,1},{20,6},{21,4},{22,4},{23,2},{24,7},{25,3},{26,3},{27,3},{28,6},{29,3},{30,7},{31,1},{32,5},{33,4},{34,3},{35,4},{36,10},{37,2},{38,3},{39,4},{40,8},{41,2},{42,8},{43,3},{44,7},{45,6},{46,4},{47,3},{48,10},{49,2},{50,7},{51,5},{52,7},{53,3},{54,9},{55,6},{56,8},{57,4},{58,6},{59,2},{60,13},{61,1},{62,3},{63,7},{64,7},{65,3},{66,9},{67,2},{68,7},{69,4},{70,9},{71,3},{72,14},{73,3},{74,5},{75,7},{76,7},{77,4},{78,8},{79,3},{80,10},{81,6},{82,5},{83,2},{84,14},{85,3},{86,5},{87,6},{88,10},{89,1},{90,13},{91,5},{92,9},{93,3},{94,6},{95,5},{96,13},{97,2},{98,5},{99,8},{100,14},{101,2},{102,11},{103,2},{104,10},{105,11},{106,6},{107,1},{108,15},{109,2},{110,12},{111,6},{112,11},{113,5},{114,9},{115,6},{116,9},{117,9},{118,6},{119,6},{120,17},{121,4},{122,3},{123,5},{124,8},{125,5},{126,14},{127,1},{128,9},{129,5},{130,9},{131,2},{132,15},{133,3},{134,5},{135,10},{136,11},{137,2},{138,9},{139,2},{140,16},{141,6},{142,6},{143,6},{144,19},{145,5},{146,6},{147,7},{148,10},{149,2},{150,14},{151,5},{152,11},{153,8},{154,10},{155,8},{156,18},{157,4},{158,5},{159,8},{160,13},{161,7},{162,16},{163,5},{164,10},{165,10},{166,8},{167,2},{168,19},{169,4},{170,7},{171,7},{172,10},{173,4},{174,11},{175,9},{176,14},{177,6},{178,5},{179,3},{180,24},{181,4},{182,11},{183,5},{184,11},{185,5},{186,8},{187,5},{188,10},{189,10},{190,10},{191,5},{192,16},{193,3},{194,7},{195,8},{196,11},{197,2},{198,17},{199,2},{200,20},{201,6},{202,4},{203,7},{204,20},{205,6},{206,5},{207,9},{208,12},{209,6},{210,22},{211,3},{212,10},{213,7},{214,4},{215,8},{216,21},{217,6},{218,5},{219,7},{220,19},{221,4},{222,13},{223,6},{224,16},{225,14},{226,10},{227,2},{228,17},{229,4},{230,12},{231,10},{232,12},{233,4},{234,16},{235,7},{236,13},{237,8},{238,11},{239,6},{240,23},{241,2},{242,8},{243,10},{244,9},{245,7},{246,12},{247,6},{248,12},{249,5},{250,11},{251,5},{252,24},{253,8},{254,3},{255,11},{256,11},{257,3},{258,11},{259,5},{260,19},{261,8},{262,6},{263,4},{264,22},{265,7},{266,8},{267,5},{268,11},{269,2},{270,22},{271,2},{272,15},{273,13},{274,7},{275,10},{276,17},{277,3},{278,5},{279,9},{280,20},{281,2},{282,13},{283,3},{284,12},{285,11},{286,13},{287,5},{288,26},{289,4},{290,11},{291,7},{292,11},{293,2},{294,14},{295,10},{296,12},{297,11},{298,7},{299,9},{300,28},{301,7},{302,8},{303,5},{304,15},{305,8},{306,19},{307,5},{308,19},{309,5},{310,14},{311,3},{312,23},{313,4},{314,9},{315,19},{316,9},{317,4},{318,14},{319,9},{320,17},{321,4},{322,11},{323,6},{324,26},{325,8},{326,9},{327,6},{328,14},{329,9},{330,22},{331,3},{332,14},{333,15},{334,4},{335,6},{336,27},{337,5},{338,8},{339,9},{340,18},{341,6},{342,16},{343,6},{344,14},{345,10},{346,10},{347,2},{348,19},{349,3},{350,20},{351,15},{352,17},{353,3},{354,14},{355,6},{356,9},{357,15},{358,6},{359,6},{360,32},{361,4},{362,8},{363,10},{364,22},{365,8},{366,9},{367,5},{368,15},{369,8},{370,12},{371,9},{372,17},{373,2},{374,9},{375,11},{376,15},{377,8},{378,23},{379,2},{380,20},{381,6},{382,7},{383,5},{384,20},{385,13},{386,9},{387,10},{388,16},{389,3},{390,19},{391,6},{392,19},{393,7},{394,6},{395,8},{396,29},{397,9},{398,4},{399,11},{400,26},{401,4},{402,14},{403,5},{404,10},{405,16},{406,14},{407,8},{408,29},{409,3},{410,11},{411,7},{412,11},{413,6},{414,17},{415,8},{416,18},{417,5},{418,12},{419,5},{420,37},{421,2},{422,9},{423,9},{424,14},{425,6},{426,15},{427,4},{428,10},{429,13},{430,13},{431,8},{432,29},{433,4},{434,11},{435,11},{436,11},{437,7},{438,13},{439,4},{440,28},{441,15},{442,10},{443,3},{444,23},{445,5},{446,9},{447,4},{448,20},{449,5},{450,26},{451,9},{452,15},{453,8},{454,6},{455,11},{456,25},{457,2},{458,7},{459,12},{460,25},{461,4},{462,22},{463,6},{464,18},{465,14},{466,8},{467,3},{468,33},{469,6},{470,14},{471,6},{472,19},{473,10},{474,14},{475,12},{476,23},{477,13},{478,10},{479,4},{480,30},{481,6},{482,7},{483,16},{484,15},{485,7},{486,25},{487,2},{488,14},{489,11},{490,16},{491,7},{492,22},{493,7},{494,13},{495,19},{496,15},{497,7},{498,14},{499,3},{500,25},{501,5},{502,8},{503,4},{504,36},{505,5},{506,16},{507,8},{508,12},{509,4},{510,23},{511,7},{512,13},{513,13},{514,7},{515,7},{516,21},{517,11},{518,11},{519,7},{520,28},{521,1},{522,19},{523,2},{524,13},{525,22},{526,9},{527,5},{528,30},{529,6},{530,13},{531,13},{532,19},{533,6},{534,13},{535,8},{536,15},{537,8},{538,5},{539,7},{540,40},{541,5},{542,7},{543,8},{544,19},{545,9},{546,25},{547,4},{548,12},{549,9},{550,19},{551,8},{552,22},{553,6},{554,6},{555,12},{556,12},{557,5},{558,20},{559,7},{560,28},{561,13},{562,6},{563,3},{564,24},{565,9},{566,9},{567,14},{568,16},{569,5},{570,22},{571,4},{572,25},{573,8},{574,13},{575,11},{576,33},{577,3},{578,12},{579,5},{580,21},{581,7},{582,17},{583,9},{584,14},{585,19},{586,8},{587,6},{588,26},{589,6},{590,16},{591,7},{592,15},{593,4},{594,26},{595,14},{596,13},{597,6},{598,15},{599,3},{600,37},{601,4},{602,12},{603,9},{604,12},{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(* Test the accuracy of the data *)

Take[TwoToNData,30]
Table[PrimeOmega[2^j-1],{j,1,30}]

(* Data looks good. Analyze M_n M_{n+1} *)

ratioData2n=Table[(TwoToNData[[j]]+TwoToNData[[j-1]])/Log[Log[(2^(j-1)-1)(2^(j)-1)]],{j,2,Length[TwoToNData]}];
Show[
ListPlot[ratioData2n,PlotRange->{All,{0,2}},PlotStyle->Directive[PointSize[0.01],Blue]]
,
Graphics[{Red,FontSize->42,Text[Subscript[β,3],{-10,b3-.15}]}],
Plot[b3,{x,0,Length[TwoToNData]+10},PlotStyle->Directive[Red,Thickness[0.005],Dotted]],AxesStyle->Thick,LabelStyle->Directive[Black,Bold,FontSize->36],PlotRange->{{-10,Length[TwoToNData]},{0,5}},ImageSize -> {958.85546875, Automatic}]