Enumeration of the prime triplets (q, q+4, q+6) to 1e16

Thomas R. Nicely

http://www.trnicely.net
Current e-mail address

Freeware copyright (c) 2010 Thomas R. Nicely. Released into the public domain by the author, who disclaims any legal liability arising from its use.

Last updated 1000 GMT 18 January 2010.

This is a table of values of pi_3b(x), the count of prime triplets (q, q+4, q+6) such that q <= x. The first three such triplets are (7, 11, 13), (13, 17, 19), and (37, 41, 43). Also provided are the values of the related functions delta_3b(x), S_3b(x), and F_3b(x). The counts in this table were obtained by a direct and explicit generation and enumeration of the primes.

The domain of this table consists of each decade from 10 through 1e12, then each 1e12 to 1e16. See also Enumeration of the prime triplets (q, q+4, q+6) from 1e16 to 2e16.

Complete counts and reciprocal sums of the prime constellations from Nicely's computations (1993-2009), including the prime triplets, are also available. These data files are very large (over 60MB each, even for the zipped versions), including more than two million data points from 0 to 2e16 at intervals of 1e10 or better.

The symbols are defined as follows.


========================================================================================
      x              pi_3b(x)    delta_3b(x)         S_3b(x)               F_3b(x)
========================================================================================
 1.000000e+01               1         7.4883  0.31068931068931068931  1.119337987868808
 2.000000e+01               2         8.0089  0.49906749597152073932  0.976800517693665
 3.000000e+01               2         8.8783  0.49906749597152073932  0.869686415572539
 4.000000e+01               3         8.5182  0.57374058085447516283  0.888807155096700
 5.000000e+01               3         9.0381  0.57374058085447516283  0.853889181630318
 6.000000e+01               3         9.4832  0.57374058085447516283  0.829494608317647
 7.000000e+01               4         8.8767  0.61644909116804334354  0.853980467620303
 8.000000e+01               4         9.2322  0.61644909116804334354  0.839724674126880
 9.000000e+01               4         9.5584  0.61644909116804334354  0.828189120283933
 1.000000e+02               5         8.8612  0.64636809748164637835  0.848530266776521
 2.000000e+02               7         9.1817  0.68987955657523238363  0.842606380489695
 3.000000e+02               9         8.8905  0.71383835614770708171  0.845623164377021
 4.000000e+02              10         9.3166  0.72350600740045011893  0.842939262830986
 5.000000e+02              11         9.5723  0.73002321581867596169  0.841033640618815
 6.000000e+02              11        10.7112  0.73002321581867596169  0.834795909081021
 7.000000e+02              12        10.7639  0.73489079161723453878  0.834790773053700
 8.000000e+02              12        11.7496  0.73489079161723453878  0.830839445187935
 9.000000e+02              15         9.6813  0.74543248681254741290  0.838087216712802
 1.000000e+03              15        10.5684  0.74543248681254741290  0.835282339832492
 2.000000e+03              26         6.9667  0.76663518342804248655  0.840844928398515
 3.000000e+03              32         6.9578  0.77403003925051776692  0.840913726738672
 4.000000e+03              37         7.2273  0.77841038101880348505  0.840734765380267
 5.000000e+03              40         9.0346  0.78042321195073799184  0.839524679004412
 6.000000e+03              46         7.5119  0.78368604565564318152  0.840336213434713
 7.000000e+03              48         9.7375  0.78458351232417617419  0.839278187206207
 8.000000e+03              51        10.7626  0.78576725056937950027  0.838848697427396
 9.000000e+03              53        12.6226  0.78647689019121002878  0.838193884814640
 1.000000e+04              57        12.3435  0.78777221336872654905  0.838312755692445
 2.000000e+04              85        16.7669  0.79342739900015808601  0.837140834307049
 3.000000e+04             111        18.3676  0.79672311440771957541  0.837065554913244
 4.000000e+04             136        18.3543  0.79888904490122440295  0.837070750879840
 5.000000e+04             161        16.6106  0.80056659264389996993  0.837189644759079
 6.000000e+04             180        19.6015  0.80160481605022549193  0.837024125876894
 7.000000e+04             193        27.6096  0.80219987240301470598  0.836647137057475
 8.000000e+04             211        29.8225  0.80291885784458232050  0.836556082245638
 9.000000e+04             231        29.3729  0.80362686070296473208  0.836573062760293
 1.000000e+05             248        31.3588  0.80416784407946939178  0.836513791166649
 2.000000e+05             424        25.2885  0.80785773656035259656  0.836634328413092
 3.000000e+05             589         9.4670  0.80987591760785123495  0.836831900322115
 4.000000e+05             749       -12.9938  0.81125588233750219889  0.837022912739692
 5.000000e+05             883       -17.3244  0.81215571008386366039  0.837053863474188
 6.000000e+05            1010       -20.4982  0.81284965760964032181  0.837070099854168
 7.000000e+05            1135       -26.2653  0.81342840288339493186  0.837097205093536
 8.000000e+05            1232        -7.7849  0.81381633346257018568  0.837022375417985
 9.000000e+05            1332         4.5431  0.81417103736666975840  0.836980068735212
 1.000000e+06            1444         2.1661  0.81452515821817359533  0.836987621473160
 2.000000e+06            2508       -62.8064  0.81676399976075195519  0.837131455740618
 3.000000e+06            3384       -43.2499  0.81783318070319361958  0.837108244572854
 4.000000e+06            4205       -28.0995  0.81854349212377251677  0.837095928366391
 5.000000e+06            5011       -38.7090  0.81908262101176086403  0.837102165738249
 6.000000e+06            5775       -37.9070  0.81949955371222070761  0.837100554776117
 7.000000e+06            6539       -61.3520  0.81985295424112827351  0.837111354417948
 8.000000e+06            7281       -82.7429  0.82015018771751519642  0.837119834819122
 9.000000e+06            7981       -78.9880  0.82039738506224501411  0.837118306934284
 1.000000e+07            8677       -85.7671  0.82061731767740791847  0.837120351905561
 2.000000e+07           15021       -66.5400  0.82194739665905949558  0.837117607438122
 3.000000e+07           20862      -116.4663  0.82266069997767088291  0.837124787242827
 4.000000e+07           26328      -125.4878  0.82313271313076116287  0.837125276920377
 5.000000e+07           31453       -25.2138  0.82347605510349102132  0.837118575924212
 6.000000e+07           36466        11.8010  0.82375009840305055202  0.837116272465644
 7.000000e+07           41422       -33.9741  0.82397937792629473493  0.837118402648468
 8.000000e+07           46193       -10.3322  0.82417054860326451337  0.837117455546580
 9.000000e+07           50969       -89.7809  0.82433937705943458946  0.837120300258024
 1.000000e+08           55556       -65.1357  0.82448431075064018795  0.837119446331570
 2.000000e+08           98418        69.2969  0.82538168403580750962  0.837117030452141
 3.000000e+08          138001        67.0714  0.82586416232420155468  0.837117026838656
 4.000000e+08          175489       139.6088  0.82618800074058826179  0.837116359567311
 5.000000e+08          211414       362.5082  0.82642867600194068560  0.837114898534859
 6.000000e+08          246548       301.4153  0.82662089175735872774  0.837115198669227
 7.000000e+08          280771       289.7789  0.82677919612592762748  0.837115245574767
 8.000000e+08          314403       153.9687  0.82691398824261155145  0.837115824636110
 9.000000e+08          347389        55.1795  0.82703059808534037348  0.837116208211721
 1.000000e+09          379748        54.7257  0.82713290960566535267  0.837116226607881
 2.000000e+09          683233       582.5740  0.82776730938276520432  0.837114859245829
 3.000000e+09          966004       193.6008  0.82811181596239448916  0.837115225513905
 4.000000e+09         1234877       779.3958  0.82834404512142738732  0.837114680378105
 5.000000e+09         1495550       469.9784  0.82851862011033864030  0.837114862404421
 6.000000e+09         1747335      2094.0144  0.82865640070090959888  0.837113985852333
 7.000000e+09         1994730      2513.1675  0.82877083427297419379  0.837113787135253
 8.000000e+09         2237535      2863.3453  0.82886813094704350466  0.837113658165985
 9.000000e+09         2476301      3274.7166  0.82895250489670877274  0.837113506185596
 1.000000e+10         2712226      3065.8419  0.82902708265860758068  0.837113569430403
 2.000000e+10         4939819      3717.4620  0.82949261362892389427  0.837113378756466
 3.000000e+10         7024265      3955.1309  0.82974658520479917809  0.837113335346757
 4.000000e+10         9020863      5572.9523  0.82991904633041493192  0.837113194293600
 5.000000e+10        10956140      7336.8702  0.83004866877003985637  0.837113079485128
 6.000000e+10        12846520      7038.7844  0.83015210480159435424  0.837113099979094
 7.000000e+10        14700428      5317.5625  0.83023785904128980968  0.837113179028244
 8.000000e+10        16520200      6095.3479  0.83031077197899181403  0.837113148819750
 9.000000e+10        18313552      6222.5978  0.83037414640531655006  0.837113142083882
 1.000000e+11        20081601      8052.8760  0.83043003934225011081  0.837113086261089
 2.000000e+11        36896157      8951.7129  0.83078131254041085282  0.837113063504337
 3.000000e+11        52723320      7395.4561  0.83097413474366469735  0.837113082543712
 4.000000e+11        67948628      7511.9157  0.83110563784364005848  0.837113080955813
 5.000000e+11        82744951     10149.2546  0.83120473737222205615  0.837113065758830
 6.000000e+11        97230660     -5017.4114  0.83128399262449968198  0.837113147930016
 7.000000e+11       111436922     -6445.2491  0.83134970520651933710  0.837113155621566
 8.000000e+11       125417438     -4427.5612  0.83140571999070731865  0.837113148059183
 9.000000e+11       139211581     -6596.3694  0.83145446805856916794  0.837113156156670
 1.000000e+12       152839134     -8567.1995  0.83149754653509660540  0.837113162348843
 2.000000e+12       282843812    -11078.6977  0.83176903380709465043  0.837113162715428
 3.000000e+12       405818201    -27727.1693  0.83191883776154675026  0.837113186322866
 4.000000e+12       524501440    -48441.8592  0.83202134079338518025  0.837113204788352
 5.000000e+12       640106557    -54154.1820  0.83209876434518541847  0.837113209501250
 6.000000e+12       753341434    -54923.4126  0.83216071748900394636  0.837113210047156
 7.000000e+12       864652734    -49909.5264  0.83221220401304271683  0.837113207750780
 8.000000e+12       974356833    -45068.1806  0.83225615709681615918  0.837113205464249
 9.000000e+12      1082691886    -51161.4412  0.83229444172791002785  0.837113207695325
 1.000000e+13      1189826966    -63860.7456  0.83232830823912088710  0.837113211654384
 1.100000e+13      1295889721    -74435.2023  0.83235863717558936809  0.837113214736948
 1.200000e+13      1400958271    -51081.1233  0.83238606531229913521  0.837113208573349
 1.300000e+13      1505162255    -33051.9115  0.83241108897606044621  0.837113204271521
 1.400000e+13      1608599784    -42897.8627  0.83243408663361360344  0.837113206437303
 1.500000e+13      1711301198    -46903.6234  0.83245534443704401554  0.837113207238819
 1.600000e+13      1813336101    -59719.3827  0.83247510064882022549  0.837113209709475
 1.700000e+13      1914744658    -73875.5331  0.83249354481768115228  0.837113212329985
 1.800000e+13      2015559392    -80230.9112  0.83251083247648172842  0.837113213384709
 1.900000e+13      2115824244    -85993.8515  0.83252709598952630402  0.837113214317664
 2.000000e+13      2215575183    -94519.2913  0.83254244596655869720  0.837113215613858
 2.100000e+13      2314818524    -82981.8769  0.83255697253105741790  0.837113213910853
 2.200000e+13      2413608349    -79278.9891  0.83257075991741662805  0.837113213424842
 2.300000e+13      2511952656    -67759.0670  0.83258387486405383678  0.837113211891138
 2.400000e+13      2609900025    -75542.5116  0.83259638087054421493  0.837113212900717
 2.500000e+13      2707440191    -72809.0034  0.83260832638673545865  0.837113212568358
 2.600000e+13      2804590334    -58849.0338  0.83261975740211294078  0.837113210905079
 2.700000e+13      2901389761    -56545.3311  0.83263071725467907004  0.837113210644182
 2.800000e+13      2997849688    -61984.9761  0.83264124143158586462  0.837113211241221
 2.900000e+13      3093972347    -63421.5841  0.83265136072737860461  0.837113211374797
 3.000000e+13      3189755723    -45888.1150  0.83266110243659370788  0.837113209595301
 3.100000e+13      3285251734    -49269.7854  0.83267049642847742069  0.837113209946674
 3.200000e+13      3380456562    -58542.2490  0.83267956438564951320  0.837113210825413
 3.300000e+13      3475373938    -66975.7606  0.83268832675599193867  0.837113211612344
 3.400000e+13      3570002606    -63528.4562  0.83269680165741933447  0.837113211305986
 3.500000e+13      3664364291    -60752.2268  0.83270500764759024609  0.837113211060958
 3.600000e+13      3758480086    -71124.9222  0.83271296166834781559  0.837113211934322
 3.700000e+13      3852336194    -72743.8433  0.83272067640769603634  0.837113212077136
 3.800000e+13      3945956621    -81975.6482  0.83272816653263318584  0.837113212820021
 3.900000e+13      4039326534    -76775.9427  0.83273544256111242421  0.837113212417352
 4.000000e+13      4132466700    -71093.9392  0.83274251691249476871  0.837113211986439
 4.100000e+13      4225389733    -71088.6602  0.83274940048173239789  0.837113211984434
 4.200000e+13      4318100116    -75123.2449  0.83275610278956399469  0.837113212273199
 4.300000e+13      4410588475    -68013.9793  0.83276263171333774642  0.837113211775042
 4.400000e+13      4502878801    -68229.7236  0.83276899686991656043  0.837113211789325
 4.500000e+13      4594966296    -65717.4597  0.83277520530448936788  0.837113211622579
 4.600000e+13      4686851184    -55694.7142  0.83278126392172036915  0.837113210967528
 4.700000e+13      4778547866    -47787.6510  0.83278718007810124204  0.837113210449033
 4.800000e+13      4870071071    -52166.6478  0.83279296072087651798  0.837113210721590
 4.900000e+13      4961403737    -47413.2013  0.83279861038074335613  0.837113210424948
 5.000000e+13      5052569093    -52589.0190  0.83280413576722257229  0.837113210746231
 5.100000e+13      5143549224    -45789.1776  0.83280954071880781301  0.837113210340181
 5.200000e+13      5234347064    -26124.2409  0.83281483009408510404  0.837113209193338
 5.300000e+13      5325001437    -28751.2410  0.83282001051676724977  0.837113209344750
 5.400000e+13      5415488745    -26552.4405  0.83282508473171520024  0.837113209219984
 5.500000e+13      5505828972    -36130.8014  0.83283005774573078958  0.837113209751973
 5.600000e+13      5596002923    -35041.0945  0.83283493215437728436  0.837113209689717
 5.700000e+13      5686036452    -46010.5901  0.83283971283379908218  0.837113210273618
 5.800000e+13      5775906818    -43288.2790  0.83284440185017436085  0.837113210129916
 5.900000e+13      5865620005    -29958.5757  0.83284900264588763128  0.837113209443143
 6.000000e+13      5955193582    -20795.4627  0.83285351907332380210  0.837113208977386
 6.100000e+13      6044628656    -14210.0371  0.83285795397871085084  0.837113208648639
 6.200000e+13      6133942263    -24636.4275  0.83286231084421362593  0.837113209156376
 6.300000e+13      6223112946    -28105.0481  0.83286659113411277322  0.837113209320051
 6.400000e+13      6312134669    -16152.1652  0.83287079697869941304  0.837113208754101
 6.500000e+13      6401031901    -10899.7486  0.83287493182509667781  0.837113208510578
 6.600000e+13      6489799002     -4437.5875  0.83287899758245613727  0.837113208216785
 6.700000e+13      6578446776     -5372.6488  0.83288299681277140456  0.837113208259118
 6.800000e+13      6666997227    -33581.6601  0.83288693246582490158  0.837113209511565
 6.900000e+13      6755391417    -28065.8998  0.83289080381570989280  0.837113209266255
 7.000000e+13      6843677277    -34759.1802  0.83289461478958507084  0.837113209556685
 7.100000e+13      6931831241    -28159.0077  0.83289836609072685624  0.837113209275638
 7.200000e+13      7019880864    -33941.9082  0.83290206054791637756  0.837113209519808
 7.300000e+13      7107804939    -29077.9065  0.83290569884631135992  0.837113209316240
 7.400000e+13      7195608291    -16622.1477  0.83290928271935338329  0.837113208810110
 7.500000e+13      7283312000    -15935.6515  0.83291281446889371461  0.837113208781018
 7.600000e+13      7370913025    -22307.1904  0.83291629536076858498  0.837113209034974
 7.700000e+13      7458396258    -19005.2826  0.83291972613063276499  0.837113208906893
 7.800000e+13      7545775605    -18357.2930  0.83292310860783159869  0.837113208881339
 7.900000e+13      7633054051    -21812.6347  0.83292644413741042422  0.837113209011621
 8.000000e+13      7720233722    -30003.0641  0.83292973398385478145  0.837113209320794
 8.100000e+13      7807294655    -21511.0646  0.83293297853388447246  0.837113209001894
 8.200000e+13      7894259389    -17459.3122  0.83293617973393888425  0.837113208853025
 8.300000e+13      7981129635    -18179.2184  0.83293933869519338064  0.837113208880124
 8.400000e+13      8067896829    -13762.5461  0.83294245612207884952  0.837113208723361
 8.500000e+13      8154582539    -24467.0927  0.83294553376004685322  0.837113209102926
 8.600000e+13      8241161969    -24219.4378  0.83294857167360661861  0.837113209095190
 8.700000e+13      8327660796    -37450.7508  0.83295157166943761210  0.837113209554062
 8.800000e+13      8414060688    -44613.6554  0.83295453398735167087  0.837113209800296
 8.900000e+13      8500374715    -57592.1466  0.83295745992046705825  0.837113210240105
 9.000000e+13      8586576052    -48403.5588  0.83296034938125519616  0.837113209929735
 9.100000e+13      8672692518    -43736.5806  0.83296320410185815319  0.837113209774391
 9.200000e+13      8758730633    -49007.3149  0.83296602505357838081  0.837113209946422
 9.300000e+13      8844681417    -54157.3806  0.83296881267751356912  0.837113210114711
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========================================================================================