Thomas R. Nicely
Last updated 1000 GMT 18 January 2010.
This is an extended table of values of pi_2(x), for 1e16 <= x <= 2e16, the counts of twin-prime pairs (q, q+2) such that q <= x. Also provided are the values of the related functions delta_2(x), S_2(x), and F_2(x); see Enumeration of the twin-prime pairs to 1e16 for an explanation of these symbols and additional notes.
Complete counts and reciprocal sums of the prime constellations from Nicely's computations (1993-2009), including the twin-prime pairs, 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.
========================================================================================
x pi_2(x) delta_2(x) S_2(x) F_2(x)
========================================================================================
1.000000e+16 10304195697298 -3142802.2329 1.83048442465833848374 1.902160583104720
1.000100e+16 10305168471081 -3151722.5198 1.83048461920336818408 1.902160583106504
1.000200e+16 10306141222705 -3143764.0801 1.83048481372451477440 1.902160583104913
1.000300e+16 10307114008949 -3175705.3430 1.83048500823313682383 1.902160583111299
1.000400e+16 10308086781792 -3199524.7376 1.83048520271963532193 1.902160583116062
1.000500e+16 10309059516199 -3190186.6934 1.83048539717901026929 1.902160583114195
1.000600e+16 10310032247694 -3183214.6397 1.83048559161836807865 1.902160583112802
1.000700e+16 10311004973518 -3175849.0061 1.83048578603716068397 1.902160583111329
1.000800e+16 10311977730242 -3204660.2223 1.83048598044270134639 1.902160583117087
1.000900e+16 10312950441802 -3193583.7181 1.83048617481979290137 1.902160583114874
1.001000e+16 10313923153873 -3188293.9235 1.83048636917756723680 1.902160583113816
1.001100e+16 10314895858809 -3181144.2684 1.83048656351450076465 1.902160583112388
1.001200e+16 10315868564158 -3179682.1831 1.83048675783210550531 1.902160583112096
1.001300e+16 10316841237579 -3151566.0978 1.83048695212392526530 1.902160583106480
1.001400e+16 10317814000098 -3217821.4429 1.83048714641413753474 1.902160583119713
1.001500e+16 10318786691494 -3218226.6489 1.83048734067074489473 1.902160583119794
1.001600e+16 10319759360269 -3201283.1465 1.83048753490343931230 1.902160583116411
1.001700e+16 10320732077184 -3237751.3663 1.83048772912635452637 1.902160583123692
1.001800e+16 10321704743953 -3229344.7394 1.83048792331986998552 1.902160583122014
1.001900e+16 10322677462301 -3277787.6967 1.83048811750429840813 1.902160583131684
1.002000e+16 10323650152092 -3302943.6692 1.83048831166364608448 1.902160583136706
1.002100e+16 10324622784263 -3275749.0883 1.83048850579211704953 1.902160583131278
1.002200e+16 10325595444585 -3281974.3852 1.83048869990683502733 1.902160583132520
1.002300e+16 10326568076309 -3264869.9914 1.83048889399647849627 1.902160583129107
1.002400e+16 10327540727087 -3272087.3386 1.83048908807056001441 1.902160583130547
1.002500e+16 10328513310680 -3217386.8583 1.83048928211187755891 1.902160583119634
1.002600e+16 10329485928355 -3202034.9824 1.83048947614063915785 1.902160583116571
1.002700e+16 10330458562372 -3208291.1430 1.83048967015330938690 1.902160583117819
1.002800e+16 10331431152708 -3176131.7719 1.83048986413791914592 1.902160583111405
1.002900e+16 10332403735969 -3142162.3014 1.83049005810177433584 1.902160583104630
1.003000e+16 10333376293959 -3088186.1638 1.83049025204125150243 1.902160583093867
1.003100e+16 10334348890174 -3077698.7914 1.83049044596901484424 1.902160583091776
1.003200e+16 10335321490430 -3076515.6169 1.83049063987825217936 1.902160583091540
1.003300e+16 10336294030999 -3020908.0728 1.83049083375626266333 1.902160583080455
1.003400e+16 10337266640692 -3039686.5919 1.83049102762872867784 1.902160583084198
1.003500e+16 10338239213306 -3026647.6071 1.83049122147448382686 1.902160583081599
1.003600e+16 10339211827834 -3060783.5513 1.83049141530927599556 1.902160583088402
1.003700e+16 10340184386243 -3044060.8578 1.83049160911357238958 1.902160583085070
1.003800e+16 10341156941291 -3029236.9597 1.83049180289789081437 1.902160583082116
1.003900e+16 10342129504866 -3028199.2904 1.83049199666460390794 1.902160583081909
1.004000e+16 10343102064744 -3028723.2834 1.83049219041128027931 1.902160583082013
1.004100e+16 10344074603688 -3013571.3721 1.83049238413449015038 1.902160583078995
1.004200e+16 10345047099294 -2960338.9904 1.83049257782977605526 1.902160583068393
1.004300e+16 10346019652332 -2969795.5721 1.83049277151721216041 1.902160583070276
1.004400e+16 10346992171548 -2950686.5510 1.83049296517862824283 1.902160583066471
1.004500e+16 10347964717243 -2963312.3613 1.83049315882603664806 1.902160583068985
1.004600e+16 10348937232914 -2951169.4372 1.83049335244819001654 1.902160583066567
1.004700e+16 10349909777116 -2972812.2128 1.83049354605675091757 1.902160583070876
1.004800e+16 10350882299179 -2977570.1226 1.83049373964163542153 1.902160583071823
1.004900e+16 10351854816533 -2982872.6012 1.83049393320631777851 1.902160583072878
1.005000e+16 10352827281889 -2941430.0832 1.83049412674139059693 1.902160583064631
1.005100e+16 10353799759726 -2917721.0033 1.83049432025969059971 1.902160583059912
1.005200e+16 10354772238387 -2900087.7964 1.83049451375890201068 1.902160583056404
1.005300e+16 10355744759910 -2930567.8975 1.83049470724739201139 1.902160583062468
1.005400e+16 10356717240641 -2925506.7418 1.83049490070852118357 1.902160583061461
1.005500e+16 10357689733474 -2937797.7644 1.83049509415281656746 1.902160583063906
1.005600e+16 10358662181620 -2910651.4007 1.83049528756898614609 1.902160583058506
1.005700e+16 10359634666988 -2925976.0861 1.83049548097332530960 1.902160583061554
1.005800e+16 10360607147059 -2941252.2563 1.83049567435738167767 1.902160583064592
1.005900e+16 10361579621105 -2955751.3468 1.83049586772101381013 1.902160583067475
1.006000e+16 10362552057661 -2938007.7937 1.83049606105797038371 1.902160583063947
1.006100e+16 10363524533698 -2964992.0326 1.83049625438355806592 1.902160583069312
1.006200e+16 10364496977791 -2965278.4998 1.83049644768358176948 1.902160583069369
1.006300e+16 10365469411049 -2959975.6314 1.83049664096224175971 1.902160583068315
1.006400e+16 10366441818598 -2934208.8636 1.83049683421658655818 1.902160583063194
1.006500e+16 10367414247866 -2935405.6329 1.83049702745604583561 1.902160583063431
1.006600e+16 10368386680155 -2944867.3757 1.83049722067690709177 1.902160583065311
1.006700e+16 10369359083413 -2930541.5287 1.83049741387280629771 1.902160583062465
1.006800e+16 10370331499377 -2934164.5287 1.83049760705203924984 1.902160583063185
1.006900e+16 10371303893841 -2921529.8124 1.83049780020781508241 1.902160583060675
1.007000e+16 10372276284245 -2910076.8169 1.83049799334360232321 1.902160583058400
1.007100e+16 10373248718285 -2947500.9793 1.83049818646887749101 1.902160583065833
1.007200e+16 10374221113628 -2951468.7367 1.83049837956729248343 1.902160583066621
1.007300e+16 10375193435927 -2887632.5266 1.83049857263203292346 1.902160583053945
1.007400e+16 10376165842943 -2913752.7863 1.83049876569442777119 1.902160583059131
1.007500e+16 10377138227395 -2922547.9535 1.83049895873317980397 1.902160583060877
1.007600e+16 10378110590285 -2915019.4657 1.83049915174849212490 1.902160583059383
1.007700e+16 10379082933934 -2893487.7608 1.83049934474083090514 1.902160583055109
1.007800e+16 10380055264119 -2863729.2767 1.83049953771134664927 1.902160583049203
1.007900e+16 10381027625666 -2870569.4514 1.83049973066893917323 1.902160583050561
1.008000e+16 10382000010764 -2906196.7231 1.83049992361206135970 1.902160583057630
1.008100e+16 10382972388070 -2939267.5300 1.83050011653449708078 1.902160583064192
1.008200e+16 10383944717073 -2929270.3105 1.83050030942821389181 1.902160583062208
1.008300e+16 10384917057407 -2935838.5031 1.83050050230504678120 1.902160583063511
1.008400e+16 10385889416123 -2966022.5464 1.83050069516639825775 1.902160583069498
1.008500e+16 10386861719778 -2946378.8791 1.83050088799770465693 1.902160583065602
1.008600e+16 10387834068204 -2976738.9400 1.83050108081876958311 1.902160583071622
1.008700e+16 10388806388031 -2983732.1681 1.83050127361504741297 1.902160583073009
1.008800e+16 10389778697274 -2985373.0025 1.83050146639011429916 1.902160583073334
1.008900e+16 10390751027085 -3012812.8824 1.83050165915015036919 1.902160583078774
1.009000e+16 10391723295899 -2984486.2470 1.83050185187899027738 1.902160583073159
1.009100e+16 10392695620809 -3017485.5359 1.83050204459984872613 1.902160583079700
1.009200e+16 10393667858169 -2968164.1884 1.83050223728425855942 1.902160583069925
1.009300e+16 10394640109384 -2937926.6443 1.83050242995232229539 1.902160583063933
1.009400e+16 10395612371818 -2924136.3433 1.83050262260352052624 1.902160583061201
1.009500e+16 10396584649294 -2930615.7253 1.83050281523861449680 1.902160583062485
1.009600e+16 10397556976945 -2992497.2303 1.83050300786456714045 1.902160583074744
1.009700e+16 10398529244560 -2999569.2984 1.83050320045954891006 1.902160583076145
1.009800e+16 10399501494686 -2994378.3699 1.83050339303199300299 1.902160583075117
1.009900e+16 10400473781536 -3031136.8851 1.83050358559264069697 1.902160583082397
1.010000e+16 10401446036499 -3041233.2843 1.83050377812790747744 1.902160583084396
1.010100e+16 10402418305756 -3070848.0083 1.83050397064694270525 1.902160583090260
1.010200e+16 10403390536975 -3067648.4978 1.83050416313938827994 1.902160583089627
1.010300e+16 10404362787246 -3088724.1934 1.83050435561655174654 1.902160583093799
1.010400e+16 10405334999328 -3076833.5361 1.83050454806710505990 1.902160583091445
1.010500e+16 10406307202302 -3061056.9670 1.83050474049680980536 1.902160583088323
1.010600e+16 10407279398308 -3043533.9272 1.83050493290609305895 1.902160583084854
1.010700e+16 10408251576906 -3013823.8580 1.83050512529289361882 1.902160583078975
1.010800e+16 10409223774697 -3008527.2007 1.83050531766445770635 1.902160583077927
1.010900e+16 10410196010669 -3046631.3969 1.83050551002454518777 1.902160583085466
1.011000e+16 10411168227993 -3071306.8881 1.83050570236191594228 1.902160583090348
1.011100e+16 10412140414138 -3070022.1162 1.83050589467409552266 1.902160583090094
1.011200e+16 10413112642902 -3116574.5228 1.83050608697568588539 1.902160583099302
1.011300e+16 10414084833682 -3130360.5500 1.83050627925074771270 1.902160583102028
1.011400e+16 10415057003893 -3128794.6399 1.83050647150272993256 1.902160583101718
1.011500e+16 10416029149802 -3108143.2347 1.83050666373089922268 1.902160583097635
1.011600e+16 10417001323569 -3120565.7765 1.83050685594557344771 1.902160583100091
1.011700e+16 10417973524071 -3164938.7079 1.83050704814653286704 1.902160583108863
1.011800e+16 10418945673199 -3163152.4714 1.83050724031833993421 1.902160583108510
1.011900e+16 10419917782683 -3126936.5097 1.83050743246331901742 1.902160583101352
1.012000e+16 10420889943226 -3146993.2654 1.83050762459940153387 1.902160583105316
1.012100e+16 10421862084534 -3153028.1815 1.83050781671269804221 1.902160583106509
1.012200e+16 10422834306745 -3245178.7009 1.83050800882300022721 1.902160583124717
1.012300e+16 10423806423273 -3236858.2668 1.83050820089344323635 1.902160583123074
1.012400e+16 10424778570521 -3264469.3223 1.83050839295098234215 1.902160583128528
1.012500e+16 10425750639065 -3218587.3109 1.83050858497400442035 1.902160583119465
1.012600e+16 10426722810576 -3280882.6759 1.83050877699840056173 1.902160583131769
1.012700e+16 10427694922119 -3288419.8609 1.83050896899199040780 1.902160583133258
1.012800e+16 10428667015687 -3283191.3096 1.83050916096307299732 1.902160583132225
1.012900e+16 10429639155669 -3329585.4658 1.83050935292436713505 1.902160583141387
1.013000e+16 10430611226698 -3312234.7734 1.83050954485309625414 1.902160583137961
1.013100e+16 10431583300400 -3302764.6764 1.83050973676340736798 1.902160583136092
1.013200e+16 10432555379956 -3304355.6190 1.83050992865593193978 1.902160583136406
1.013300e+16 10433527478490 -3330131.0453 1.83051012053326446102 1.902160583141493
1.013400e+16 10434499543430 -3327518.3998 1.83051031238503175987 1.902160583140978
1.013500e+16 10435471641721 -3363462.1269 1.83051050422445009488 1.902160583148072
1.013600e+16 10436443718475 -3383073.6712 1.83051069604069108724 1.902160583151942
1.013700e+16 10437415764321 -3376981.4774 1.83051088783191047126 1.902160583150740
1.013800e+16 10438387793500 -3359425.9904 1.83051107960092260872 1.902160583147276
1.013900e+16 10439359854980 -3379374.6551 1.83051127135739180441 1.902160583151212
1.014000e+16 10440331897392 -3385457.9165 1.83051146309118796532 1.902160583152412
1.014100e+16 10441303918209 -3375148.2198 1.83051165480181703752 1.902160583150378
1.014200e+16 10442275914057 -3345071.0102 1.83051184648861858414 1.902160583144447
1.014300e+16 10443247958417 -3368706.7332 1.83051203816608700088 1.902160583149107
1.014400e+16 10444219955824 -3350589.8342 1.83051222981540100780 1.902160583145535
1.014500e+16 10445191978472 -3362913.7590 1.83051242145079894998 1.902160583147965
1.014600e+16 10446163964616 -3343932.9531 1.83051261306011230893 1.902160583144223
1.014700e+16 10447136013848 -3393238.8624 1.83051280466297685610 1.902160583153942
1.014800e+16 10448107965262 -3349924.9330 1.83051299622768032511 1.902160583145405
1.014900e+16 10449079905846 -3300978.6108 1.83051318777137354416 1.902160583135759
1.015000e+16 10450051905813 -3316612.3422 1.83051337930789589050 1.902160583138839
1.015100e+16 10451023895559 -3327221.5732 1.83051357082353503983 1.902160583140930
1.015200e+16 10451995846532 -3304253.7505 1.83051376231266964942 1.902160583136405
1.015300e+16 10452967823920 -3312896.3204 1.83051395378814626385 1.902160583138107
1.015400e+16 10453939748646 -3274071.7297 1.83051414523439182589 1.902160583130460
1.015500e+16 10454911702465 -3269534.4251 1.83051433666751405991 1.902160583129566
1.015600e+16 10455883664146 -3278052.8535 1.83051452808333446461 1.902160583131244
1.015700e+16 10456855546538 -3212475.4618 1.83051471946469480105 1.902160583118331
1.015800e+16 10457827515830 -3238990.6971 1.83051491084432452929 1.902160583123552
1.015900e+16 10458799459654 -3245230.0067 1.83051510220010038279 1.902160583124780
1.016000e+16 10459771392843 -3246025.8379 1.83051529353494762590 1.902160583124937
1.016100e+16 10460743315671 -3241651.6380 1.83051548484892391028 1.902160583124076
1.016200e+16 10461715189497 -3193465.8548 1.83051567613442811730 1.902160583114591
1.016300e+16 10462687105955 -3193101.9357 1.83051586740949997005 1.902160583114520
1.016400e+16 10463659003804 -3179318.3286 1.83051605866209006228 1.902160583111807
1.016500e+16 10464630892905 -3161975.4814 1.83051624989414325394 1.902160583108395
1.016600e+16 10465602798823 -3166637.8421 1.83051644111069297351 1.902160583109312
1.016700e+16 10466574694394 -3166140.8588 1.83051663230639901619 1.902160583109215
1.016800e+16 10467546546157 -3127022.9797 1.83051682347468310378 1.902160583101520
1.016900e+16 10468518443504 -3138675.6531 1.83051701463313299991 1.902160583103812
1.017000e+16 10469490317380 -3132043.3276 1.83051720576816960052 1.902160583102508
1.017100e+16 10470462198768 -3138108.4516 1.83051739688589033899 1.902160583103701
1.017200e+16 10471434066808 -3136010.4739 1.83051758798219653258 1.902160583103288
1.017300e+16 10472405927493 -3131741.8432 1.83051777905827157027 1.902160583102449
1.017400e+16 10473377810500 -3154979.0085 1.83051797011995293514 1.902160583107017
1.017500e+16 10474349658720 -3148612.4188 1.83051816115601791282 1.902160583105766
1.017600e+16 10475321474496 -3114984.5232 1.83051835216693148579 1.902160583099156
1.017700e+16 10476293323080 -3119346.7709 1.83051854316552345943 1.902160583100013
1.017800e+16 10477265190863 -3148089.6113 1.83051873414912141517 1.902160583105662
1.017900e+16 10478236996822 -3120189.4938 1.83051892510180816432 1.902160583100180
1.018000e+16 10479208803863 -3098551.8682 1.83051911603594864804 1.902160583095928
1.018100e+16 10480180610563 -3081753.1839 1.83051930695126742259 1.902160583092628
1.018200e+16 10481152407756 -3060626.8910 1.83051949784596739556 1.902160583088478
1.018300e+16 10482124227059 -3066789.4392 1.83051968872626282398 1.902160583089689
1.018400e+16 10483096015716 -3047484.2786 1.83051987958179549710 1.902160583085898
1.018500e+16 10484067815568 -3044551.8593 1.83052007042078678562 1.902160583085322
1.018600e+16 10485039601628 -3033004.6316 1.83052026123833331572 1.902160583083054
1.018700e+16 10486011369666 -3008612.0460 1.83052045203360925859 1.902160583078265
1.018800e+16 10486983134516 -2986207.5527 1.83052064280953096490 1.902160583073867
1.018900e+16 10487954905008 -2974620.6025 1.83052083356783554461 1.902160583071592
1.019000e+16 10488926693522 -2986230.6461 1.83052102431095643470 1.902160583073871
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