FINDING THE PRIMES BY A FACTORIAL TREE
The naturals (n=0,1,2,3,...) can be divided into a binary tree by a self similar process as following;
(n)
(2n,2n+1)
((4n,4n+2),(4n+1,4n+3))
(((8n,8n+4),(8n+2,8n+6)),((8n+1,8n+5),(8n+3,8n+7)))
((((16n,16n+8),(16n+4,16n+12)),((16n+2,16n+10),(16n+6,16n+14))),
(((16n+1,16n+9),(16n+5,16n+13)),((16n+3,16n+11),(16n+7,16n+15))))
...
But this method stops being useful to find the primes at the second step. After that, the primes are divided statistically equally among the odd sets due to the Dirichlet's theorem.
The naturals can be divided into a factorial tree instead of the binary one by a similar self similar process, producing a tree which sorts the primes into tightly concentrated groups. The efficiency of this tree increases by the depth of the tree. This can be proven.
Following demonstrates the sorting of the primes for the factorial tree of depth 6.
(#)... sets of the form an+b containing the primes, where a,b are coprimes.
(##)... a,b coprime; b prime...only for depth 6
(n)

(2n,2n+1#)

((6n,6n+2,6n+4),
(6n+1#,6n+3,6n+5#))

(((24n,24n+6,24n+12,24n+18),
(24n+2,24n+8,24n+14,24n+20),
(24n+4,24n+10,24n+16,24n+22)),
((24n+1#,24n+7#,24n+13#,24n+19#),
(24n+3,24n+9,24n+15,24n+21),
(24n+5#,24n+11#,24n+17#,24n+23#)))

((((120n,120n+24,120n+48,120n+72,120n+96),
(120n+6,120n+30,120n+54,120n+78,120n+102),
(120n+12,120n+36,120n+60,120n+84,120n+108),
(120n+18,120n+42,120n+66,120n+90,120n+114)),
((120n+2,120n+26,120n+50,120n+74,120n+98),
(120n+8,120n+32,120n+56,120n+80,120n+104),
(120n+14,120n+38,120n+62,120n+86,120n+110),
(120n+20,120n+44,120n+68,120n+92,120n+116)),
((120n+4,120n+28,120n+52,120n+76,120n+100),
(120n+10,120n+34,120n+58,120n+82,120n+106),
(120n+16,120n+40,120n+64,120n+88,120n+112),
(120n+22,120n+46,120n+70,120n+94,120n+118))),
(((120n+1#,120n+25,120n+49#,120n+73#,120n+97#),
(120n+7#,120n+31#,120n+55,120n+79#,120n+103#),
(120n+13#,120n+37#,120n+61#,120n+85,120n+109#),
(120n+19#,120n+43#,120n+67#,120n+91#,120n+115)),
((120n+3,120n+27,120n+51,120n+75,120n+99),
(120n+9,120n+33,120n+57,120n+81,120n+105),
(120n+15,120n+39,120n+63,120n+87,120n+111),
(120n+21,120n+45,120n+69,120n+93,120n+117)),
((120n+5,120n+29#,120n+53#,120n+77#,120n+101#),
(120n+11#,120n+35,120n+59#,120n+83#,120n+107#),
(120n+17#,120n+41#,120n+65,120n+89#,120n+113#),
(120n+23#,120n+47#,120n+71#,120n+95,120n+119#))))

(((((720n,720n+120,720n+240,720n+360,720n+480,720n+600),
(720n+24,720n+144,720n+264,720n+384,720n+504,720n+624),
(720n+48,720n+168,720n+288,720n+408,720n+528,720n+648),
(720n+72,720n+192,720n+312,720n+432,720n+552,720n+672),
(720n+96,720n+216,720n+336,720n+456,720n+576,720n+696)),
((720n+6,720n+126,720n+246,720n+366,720n+486,720n+606),
(720n+30,720n+150,720n+270,720n+390,720n+510,720n+630),
(720n+54,720n+174,720n+294,720n+414,720n+534,720n+654),
(720n+78,720n+198,720n+318,720n+438,720n+558,720n+678),
(720n+102,720n+222,720n+342,720n+462,720n+582,720n+702)),
((720n+12,720n+132,720n+252,720n+372,720n+492,720n+612),
(720n+36,720n+156,720n+276,720n+396,720n+516,720n+636),
(720n+60,720n+180,720n+300,720n+420,720n+540,720n+660),
(720n+84,720n+204,720n+324,720n+444,720n+564,720n+684),
(720n+108,720n+228,720n+348,720n+468,720n+588,720n+708)),
((720n+18,720n+138,720n+258,720n+378,720n+498,720n+618),
(720n+42,720n+162,720n+282,720n+402,720n+522,720n+642),
(720n+72,720n+186,720n+306,720n+426,720n+546,720n+666),
(720n+90,720n+210,720n+330,720n+450,720n+570,720n+690),
(720n+114,720n+234,720n+354,720n+474,720n+594,720n+714))),

(((720n+2,720n+122,720n+242,720n+362,720n+482,720n+602),
(720n+26,720n+146,720n+266,720n+386,720n+506,720n+626),
(720n+50,720n+170,720n+290,720n+410,720n+530,720n+650),
(720n+74,720n+194,720n+314,720n+434,720n+554,720n+674),
(720n+98,720n+218,720n+338,720n+458,720n+578,720n+698)),
((720n+8,720n+128,720n+248,720n+368,720n+488,720n+608),
(720n+32,720n+152,720n+272,720n+392,720n+512,720n+632),
(720n+56,720n+176,720n+296,720n+416,720n+536,720n+656),
(720n+80,720n+200,720n+320,720n+440,720n+560,720n+680),
(720n+104,720n+224,720n+344,720n+464,720n+584,720n+704)),
((720n+14,720n+134,720n+254,720n+374,720n+494,720n+614),
(720n+38,720n+158,720n+278,720n+398,720n+518,720n+638),
(720n+62,720n+182,720n+302,720n+422,720n+542,720n+662),
(720n+86,720n+206,720n+326,720n+446,720n+566,720n+686),
(720n+110,720n+230,720n+350,720n+470,720n+590,720n+710)),
((720n+20,720n+140,720n+260,720n+380,720n+500,720n+620),
(720n+44,720n+164,720n+284,720n+404,720n+524,720n+644),
(720n+68,720n+188,720n+308,720n+428,720n+548,720n+668),
(720n+92,720n+212,720n+332,720n+452,720n+572,720n+692),
(720n+116,720n+236,720n+356,720n+476,720n+596,720n+716))),

(((720n+4,720n+124,720n+244,720n+364,720n+484,720n+604),
(720n+28,720n+148,720n+268,720n+388,720n+508,720n+628),
(720n+52,720n+172,720n+292,720n+412,720n+532,720n+652),
(720n+76,720n+196,720n+316,720n+436,720n+556,720n+676),
(720n+100,720n+220,720n+340,720n+460,720n+580,720n+700)),
((720n+10,720n+130,720n+250,720n+370,720n+490,720n+610),
(720n+34,720n+154,720n+274,720n+394,720n+514,720n+634),
(720n+58,720n+178,720n+298,720n+418,720n+538,720n+658),
(720n+82,720n+202,720n+322,720n+442,720n+562,720n+682),
(720n+106,720n+226,720n+346,720n+466,720n+586,720n+706)),
((720n+16,720n+136,720n+256,720n+376,720n+496,720n+616),
(720n+40,720n+160,720n+280,720n+400,720n+520,720n+640),
(720n+64,720n+184,720n+304,720n+424,720n+544,720n+664),
(720n+88,720n+208,720n+328,720n+448,720n+568,720n+688),
(720n+112,720n+232,720n+352,720n+472,720n+592,720n+712)),
((720n+22,720n+142,720n+262,720n+382,720n+502,720n+622),
(720n+46,720n+166,720n+286,720n+406,720n+526,720n+646),
(720n+70,720n+190,720n+310,720n+430,720n+550,720n+670),
(720n+94,720n+214,720n+334,720n+454,720n+574,720n+694),
(720n+118,720n+238,720n+358,720n+478,720n+598,720n+718)))),

((((720n+1#,720n+121#,720n+241##,720n+361#,720n+481#,720n+601##),
(720n+25,720n+145,720n+265,720n+385,720n+505,720n+625),
(720n+49#,720n+169#,720n+289#,720n+409##,720n+529#,720n+649#),
(720n+73##,720n+193##,720n+313##,720n+433##,720n+553#,720n+673##),
(720n+97##,720n+217#,720n+337##,720n+457##,720n+577##,720n+697#)),
((720n+7##,720n+127##,720n+247#,720n+367##,720n+487##,720n+607##),
(720n+31##,720n+151##,720n+271##,720n+391#,720n+511#,720n+631##),
(720n+55,720n+175,720n+295,720n+415,720n+535,720n+655),
(720n+79##,720n+199##,720n+319#,720n+439##,720n+559#,720n+679#),
(720n+103##,720n+223##,720n+343#,720n+463##,720n+583#,720n+703#)),
((720n+13##,720n+133#,720n+253#,720n+373##,720n+493#,720n+613##),
(720n+37##,720n+157##,720n+277##,720n+397##,720n+517#,720n+637#),
(720n+61##,720n+181##,720n+301#,720n+421##,720n+541##,720n+661##),
(720n+85,720n+205,720n+325,720n+445,720n+565,720n+685),
(720n+109##,720n+229##,720n+349##,720n+469#,720n+589#,720n+709##)),
((720n+19##,720n+139##,720n+259#,720n+379##,720n+499##,720n+619##),
(720n+43##,720n+163##,720n+283##,720n+403#,720n+523#,720n+643##),
(720n+73##,720n+187#,720n+307##,720n+427#,720n+547##,720n+667#),
(720n+91#,720n+211##,720n+331##,720n+451#,720n+571##,720n+691##),
(720n+115,720n+235,720n+355,720n+475,720n+595,720n+715))),

(((720n+3,720n+123,720n+243,720n+363,720n+483,720n+603),
(720n+27,720n+147,720n+267,720n+387,720n+507,720n+627),
(720n+51,720n+171,720n+291,720n+411,720n+531,720n+651),
(720n+75,720n+195,720n+315,720n+435,720n+555,720n+675),
(720n+99,720n+219,720n+339,720n+459,720n+579,720n+699)),
((720n+9,720n+129,720n+249,720n+369,720n+489,720n+609),
(720n+33,720n+153,720n+273,720n+393,720n+513,720n+633),
(720n+57,720n+177,720n+297,720n+417,720n+537,720n+657),
(720n+81,720n+201,720n+321,720n+441,720n+561,720n+681),
(720n+105,720n+225,720n+345,720n+465,720n+585,720n+705)),
((720n+15,720n+135,720n+255,720n+375,720n+495,720n+615),
(720n+39,720n+159,720n+279,720n+399,720n+519,720n+639),
(720n+63,720n+183,720n+303,720n+423,720n+543,720n+663),
(720n+87,720n+207,720n+327,720n+447,720n+567,720n+687),
(720n+111,720n+231,720n+351,720n+471,720n+591,720n+711)),
((720n+21,720n+141,720n+261,720n+381,720n+501,720n+621),
(720n+45,720n+165,720n+285,720n+405,720n+525,720n+645),
(720n+69,720n+189,720n+309,720n+429,720n+549,720n+669),
(720n+93,720n+213,720n+333,720n+453,720n+573,720n+693),
(720n+117,720n+237,720n+357,720n+477,720n+597,720n+717))),

(((720n+5,720n+125,720n+245,720n+365,720n+485,720n+605),
(720n+29##,720n+149##,720n+269##,720n+389##,720n+509##,720n+629#),
(720n+53##,720n+173##,720n+293##,720n+413#,720n+533#,720n+653##),
(720n+77#,720n+197##,720n+317##,720n+437#,720n+557##,720n+677##),
(720n+101##,720n+221#,720n+341#,720n+461##,720n+581#,720n+701##)),
((720n+11##,720n+131##,720n+251##,720n+371#,720n+491##,720n+611#),
(720n+35,720n+155,720n+275,720n+395,720n+515,720n+635),
(720n+59##,720n+179##,720n+299#,720n+419##,720n+539#,720n+659##),
(720n+83##,720n+203#,720n+323#,720n+443##,720n+563##,720n+683##),
(720n+107##,720n+227##,720n+347##,720n+467##,720n+587##,720n+707#)),
((720n+17##,720n+137##,720n+257##,720n+377#,720n+497#,720n+617##),
(720n+41##,720n+161#,720n+281##,720n+401##,720n+521##,720n+641##),
(720n+65,720n+185,720n+305,720n+425,720n+545,720n+665),
(720n+89##,720n+209#,720n+329#,720n+449##,720n+569##,720n+689#),
(720n+113##,720n+233##,720n+353##,720n+473#,720n+593##,720n+713#)),
((720n+23##,720n+143#,720n+263##,720n+383##,720n+503##,720n+623#),
(720n+47##,720n+167##,720n+287#,720n+407#,720n+527#,720n+647##),
(720n+71##,720n+191##,720n+311##,720n+431##,720n+551#,720n+671#),
(720n+95,720n+215,720n+335,720n+455,720n+575,720n+695),
(720n+119#,720n+239##,720n+359##,720n+479##,720n+599##,720n+719##)))))