Last updated 0400 EDT 23 July 2007.

• This is an extended table of the values of pi_2(x), the number of twin-prime pairs <= x. The counts in this table were obtained by the author, employing a direct and explicit generation and enumeration of the twin primes.
• The domain of this table is (5e15)(1e12)(9.2e15).
• See also on this site Counts of twin-prime pairs and Brun's constant to 5e15, and my most recent cumulative count of the twin primes and Brun's constant.
• The most extensive counts of pi_2(x) of which I am aware have been compiled by Tomás Oliveira e Silva (but do not include the values of the related functions delta_2(x), S_2(x), and F_2(x)).
• The full results shown for 1e16 are due to Pascal Sebah (posting of 12 December 2002).
• Also given are the values of the related functions delta_2(x), S_2(x), and F_2(x); see Counts of twin-prime pairs and Brun's constant to 5e15 for a brief description of these functions, or the papers A new error analysis for Brun's constant and Enumeration to 1.6e15 of the twin primes and Brun's constant for detailed discussions.

===============================================================================
   x          pi_2(x)      delta_2(x)          S_2(x)              F_2(x)
===============================================================================
5.000e15   5357875276068 -1320734.6414  1.8291100235436749017  1.9021605825820
5.001e15   5358885728926 -1348094.5251  1.8291104276844048603  1.9021605825929
5.002e15   5359896157931 -1362779.9668  1.8291108317347932438  1.9021605825988
5.003e15   5360906561152 -1362857.5463  1.8291112356941029614  1.9021605825988
5.004e15   5361916960765 -1370500.8444  1.8291116395712342994  1.9021605826019
5.005e15   5362927333509 -1362446.4428  1.8291120433569258893  1.9021605825986
5.006e15   5363937702938 -1362245.9243  1.8291124470606253239  1.9021605825986
5.007e15   5364948082220 -1383064.8726  1.8291128506876235809  1.9021605826069
5.008e15   5365958430736 -1384281.8725  1.8291132542217300725  1.9021605826074
5.009e15   5366968714352 -1331760.5098  1.8291136576493510144  1.9021605825864
5.010e15   5367979081719 -1374149.3712  1.8291140610298767046  1.9021605826033
5.011e15   5368989413425 -1392034.0446  1.8291144643156602922  1.9021605826105
5.012e15   5369999712832 -1388774.1188  1.8291148675080825877  1.9021605826092
5.013e15   5371010030872 -1415299.1834  1.8291152706275001575  1.9021605826197
5.014e15   5372020264920 -1368981.8294  1.8291156736330052257  1.9021605826013
5.015e15   5373030565545 -1400388.6485  1.8291160765846970226  1.9021605826138
5.016e15   5374040790275 -1367045.2335  1.8291164794257835255  1.9021605826005
5.017e15   5375051008692 -1338531.1782  1.8291168821840504104  1.9021605825891
5.018e15   5376061297490 -1391538.0774  1.8291172848901007175  1.9021605826103
5.019e15   5377071542255 -1411649.5270  1.8291176874983570339  1.9021605826183
5.020e15   5378081684373 -1340249.1236  1.8291180899855069840  1.9021605825898
5.021e15   5379091890691 -1344181.4651  1.8291184924180623486  1.9021605825914
5.022e15   5380102069190 -1331425.1504  1.8291188947593950328  1.9021605825863
5.023e15   5381112256263 -1338370.7791  1.8291192970240345149  1.9021605825891
5.024e15   5382122456184 -1369289.9520  1.8291196992137126691  1.9021605826014
5.025e15   5383132595468 -1350695.2710  1.8291201012992081622  1.9021605825940
5.026e15   5384142733587 -1342056.3389  1.8291205033042311921  1.9021605825905
5.027e15   5385152873606 -1346435.7593  1.8291209052300342375  1.9021605825923
5.028e15   5386162995630 -1343936.1370  1.8291213070687316060  1.9021605825913
5.029e15   5387173135512 -1370408.0779  1.8291217088346208555  1.9021605826018
5.030e15   5388183199228 -1331825.1887  1.8291221104903389495  1.9021605825865
5.031e15   5389193337835 -1379242.0770  1.8291225120959886462  1.9021605826053
5.032e15   5390203415388 -1376711.3517  1.8291229135975515063  1.9021605826043
5.033e15   5391213483886 -1376229.6225  1.8291233150157342670  1.9021605826041
5.034e15   5392223540325 -1374790.5000  1.8291237163493761642  1.9021605826036
5.035e15   5393233580335 -1368021.5960  1.8291241175967745083  1.9021605826009
5.036e15   5394243584668 -1336672.5232  1.8291245187503185902  1.9021605825884
5.037e15   5395253633571 -1360987.8953  1.8291249198419120294  1.9021605825981
5.038e15   5396263674932 -1388853.3268  1.8291253208508899365  1.9021605826091
5.039e15   5397273636915 -1348430.4336  1.8291257217487703896  1.9021605825931
5.040e15   5398283622694 -1342890.8322  1.8291261225765437393  1.9021605825909
5.041e15   5399293586542 -1326505.1403  1.8291265233160939469  1.9021605825844
5.042e15   5400303592370 -1363181.9764  1.8291269239928092863  1.9021605825989
5.043e15   5401313563677 -1376417.9603  1.8291273245763727072  1.9021605826042
5.044e15   5402323524507 -1390254.7123  1.8291277250763565401  1.9021605826097
5.045e15   5403333468049 -1397878.8542  1.8291281254900915446  1.9021605826127
5.046e15   5404343379890 -1384875.0085  1.8291285258119018206  1.9021605826075
5.047e15   5405353295218 -1386428.7987  1.8291289260557662292  1.9021605826082
5.048e15   5406363201573 -1390077.8494  1.8291293262167796516  1.9021605826096
5.049e15   5407373114555 -1411419.7859  1.8291297263011554515  1.9021605826181
5.050e15   5408382985094 -1401382.2349  1.8291301262894873484  1.9021605826141
5.051e15   5409392843125 -1389897.8237  1.8291305261936685160  1.9021605826095
5.052e15   5410402711718 -1400034.1808  1.8291309260228664720  1.9021605826135
5.053e15   5411412590584 -1431499.9356  1.8291313257769957283  1.9021605826260
5.054e15   5412422399367 -1403936.7185  1.8291317254242843244  1.9021605826151
5.055e15   5413432194815 -1374090.1609  1.8291321249872280789  1.9021605826033
5.056e15   5414442030740 -1395769.8951  1.8291325244871498003  1.9021605826119
5.057e15   5415451877856 -1439687.5545  1.8291329239124904315  1.9021605826292
5.058e15   5416461680632 -1450309.7733  1.8291333232413202667  1.9021605826334
5.059e15   5417471488781 -1477347.1868  1.8291337224933329737  1.9021605826441
5.060e15   5418481281879 -1500373.4313  1.8291341216604849126  1.9021605826532
5.061e15   5419491029345 -1488805.1439  1.8291345207307223537  1.9021605826487
5.062e15   5420500810672 -1522132.9630  1.8291349197354954719  1.9021605826618
5.063e15   5421510519655 -1494149.5276  1.8291353186328736195  1.9021605826508
5.064e15   5422520273428 -1521986.4780  1.8291357174691635108  1.9021605826618
5.065e15   5423529951095 -1484745.4551  1.8291361161966462862  1.9021605826471
5.066e15   5424539632325 -1462093.1012  1.8291365148468228047  1.9021605826381
5.067e15   5425549327735 -1464644.0592  1.8291369134239122754  1.9021605826391
5.068e15   5426559010461 -1465531.9732  1.8291373119173423650  1.9021605826395
5.069e15   5427568722795 -1507046.4882  1.8291377103438339337  1.9021605826558
5.070e15   5428578418605 -1543053.2501  1.8291381086852134042  1.9021605826701
5.071e15   5429588095752 -1571410.9059  1.8291385069406726723  1.9021605826812
5.072e15   5430597745317 -1583198.1035  1.8291389051067237120  1.9021605826859
5.073e15   5431607363570 -1574682.4917  1.8291393031819348072  1.9021605826825
5.074e15   5432616939570 -1534920.7203  1.8291397011620288404  1.9021605826669
5.075e15   5433626558040 -1548633.4403  1.8291400990804341441  1.9021605826723
5.076e15   5434636156296 -1553134.3033  1.8291404969124742073  1.9021605826740
5.077e15   5435645696460 -1510542.9620  1.8291408946432606769  1.9021605826573
5.078e15   5436655241137 -1483462.0702  1.8291412922974922847  1.9021605826466
5.079e15   5437664793158 -1474720.2825  1.8291416898763147745  1.9021605826431
5.080e15   5438674330584 -1462376.2545  1.8291420873711192374  1.9021605826383
5.081e15   5439683829548 -1422560.6428  1.8291424847725435576  1.9021605826226
5.082e15   5440693342567 -1407788.1050  1.8291428821012935875  1.9021605826168
5.083e15   5441702859032 -1407447.2995  1.8291432793532241077  1.9021605826167
5.084e15   5442712348102 -1390694.8859  1.8291436765162312017  1.9021605826101
5.085e15   5443721851974 -1399725.5244  1.8291440736069486136  1.9021605826136
5.086e15   5444731332052 -1395940.8766  1.8291444706102262358  1.9021605826121
5.087e15   5445740833132 -1424134.6048  1.8291448675437098590  1.9021605826232
5.088e15   5446750278472 -1407562.3722  1.8291452643772603855  1.9021605826167
5.089e15   5447759718995 -1397144.8431  1.8291456611309312613  1.9021605826126
5.090e15   5448769157028 -1395206.6828  1.8291460578056684141  1.9021605826118
5.091e15   5449778571617 -1380791.5574  1.8291464543932698865  1.9021605826062
5.092e15   5450787986209 -1377344.1340  1.8291468509029798619  1.9021605826048
5.093e15   5451797407496 -1391554.0807  1.8291472473374583256  1.9021605826104
5.094e15   5452806778591 -1366532.0666  1.8291476436743972237  1.9021605826006
5.095e15   5453816160235 -1363016.7616  1.8291480399376799718  1.9021605825992
5.096e15   5454825544184 -1372761.8367  1.8291484361241012364  1.9021605826030
5.097e15   5455834937138 -1402464.9637  1.8291488322363190324  1.9021605826147
5.098e15   5456844278459 -1391485.8156  1.8291492282505709528  1.9021605826104
5.099e15   5457853645143 -1416818.0661  1.8291496241971001235  1.9021605826203
5.100e15   5458862981039 -1422308.3901  1.8291500200539095739  1.9021605826225
5.101e15   5459872314858 -1436665.4630  1.8291504158322934602  1.9021605826281
5.102e15   5460881583953 -1397239.9618  1.8291508115077217864  1.9021605826126
5.103e15   5461890932146 -1447851.5638  1.8291512071366078291  1.9021605826325
5.104e15   5462900192734 -1421794.9478  1.8291516026536413454  1.9021605826223
5.105e15   5463909483212 -1436562.7932  1.8291519981049017714  1.9021605826281
5.106e15   5464918750041 -1438613.7804  1.8291523934694430709  1.9021605826289
5.107e15   5465928039939 -1474663.5908  1.8291527887655950493  1.9021605826430
5.108e15   5466937284354 -1476157.9068  1.8291531839665420022  1.9021605826436
5.109e15   5467946478252 -1438060.4116  1.8291535790703498613  1.9021605826287
5.110e15   5468955715817 -1454552.7896  1.8291539741139233348  1.9021605826351
5.111e15   5469964926001 -1454584.7258  1.8291543690694811850  1.9021605826351
5.112e15   5470974133178 -1462527.9065  1.8291547639465932590  1.9021605826382
5.113e15   5471983283995 -1425027.0186  1.8291551587244203691  1.9021605826236
5.114e15   5472992481329 -1444956.7501  1.8291555534432381816  1.9021605826314
5.115e15   5474001635004 -1432138.7901  1.8291559480678075274  1.9021605826263
5.116e15   5475010773318 -1414868.8284  1.8291563426092275814  1.9021605826196
5.117e15   5476019931835 -1428708.5559  1.8291567370814328656  1.9021605826250
5.118e15   5477029039105 -1402205.6642  1.8291571314565277934  1.9021605826146
5.119e15   5478038107554 -1347783.8463  1.8291575257394045022  1.9021605825934
5.120e15   5479047209249 -1337507.7956  1.8291579199582530654  1.9021605825894
5.121e15   5480056341853 -1369038.2068  1.8291583141121859695  1.9021605826017
5.122e15   5481065418924 -1355930.7755  1.8291587081674721138  1.9021605825966
5.123e15   5482074514878 -1372599.1980  1.8291591021532052799  1.9021605826031
5.124e15   5483083581438 -1370764.1719  1.8291594960505654259  1.9021605826023
5.125e15   5484092688465 -1420284.3954  1.8291598898868540276  1.9021605826217
5.126e15   5485101742167 -1427365.5679  1.8291602836254964794  1.9021605826244
5.127e15   5486110821784 -1471245.3895  1.8291606772974441507  1.9021605826416
5.128e15   5487119799179 -1423784.5614  1.8291610708527430662  1.9021605826230
5.129e15   5488128753094 -1363722.7858  1.8291614643221470504  1.9021605825996
5.130e15   5489137796488 -1404016.7656  1.8291618577497315827  1.9021605826153
5.131e15   5490146783365 -1398668.2047  1.8291622510786010054  1.9021605826132
5.132e15   5491155753993 -1387942.8081  1.8291626443244867176  1.9021605826091
5.133e15   5492164700871 -1364337.2817  1.8291630374844990424  1.9021605825999
5.134e15   5493173644994 -1348844.3320  1.8291634305668508018  1.9021605825938
5.135e15   5494182627465 -1382564.6670  1.8291638235875829729  1.9021605826070
5.136e15   5495191586692 -1403903.9950  1.8291642165227326074  1.9021605826153
5.137e15   5496200518678 -1408863.0258  1.8291646093707757645  1.9021605826172
5.138e15   5497209453693 -1427709.4697  1.8291650021435317673  1.9021605826245
5.139e15   5498218364467 -1433171.0382  1.8291653948304165452  1.9021605826267
5.140e15   5499227242739 -1416984.4437  1.8291657874282480176  1.9021605826204
5.141e15   5500236143902 -1434540.3993  1.8291661799586108489  1.9021605826272
5.142e15   5501245026218 -1444098.6192  1.8291665724052975907  1.9021605826309
5.143e15   5502253875009 -1430978.8187  1.8291669647626323067  1.9021605826258
5.144e15   5503262704966 -1409869.7136  1.8291673570363602121  1.9021605826176
5.145e15   5504271553742 -1418422.0211  1.8291677492411540834  1.9021605826209
5.146e15   5505280386311 -1421607.4589  1.8291681413634248748  1.9021605826222
5.147e15   5506289214328 -1431078.7460  1.8291685334077353688  1.9021605826259
5.148e15   5507298042910 -1451950.6020  1.8291689253761027729  1.9021605826340
5.149e15   5508306865382 -1477545.7475  1.8291693172659645892  1.9021605826439
5.150e15   5509315688792 -1514909.9043  1.8291697090800879198  1.9021605826584
5.151e15   5510324446329 -1497229.7948  1.8291701007925581541  1.9021605826516
5.152e15   5511333187880 -1474390.1425  1.8291704924227838641  1.9021605826427
5.153e15   5512341931114 -1464057.6716  1.8291708839776557504  1.9021605826387
5.154e15   5513350707873 -1498072.1075  1.8291712754695593054  1.9021605826519
5.155e15   5514359421938 -1480212.1764  1.8291716668611857893  1.9021605826449
5.156e15   5515368166033 -1503199.6053  1.8291720581885461345  1.9021605826539
5.157e15   5516376899665 -1526539.1224  1.8291724494359549744  1.9021605826629
5.158e15   5517385563137 -1490531.4565  1.8291728405802973145  1.9021605826490
5.159e15   5518394216448 -1455173.3376  1.8291732316448755216  1.9021605826352
5.160e15   5519402917771 -1478635.4964  1.8291736226522703375  1.9021605826443
5.161e15   5520411569364 -1463173.6646  1.8291740135646219597  1.9021605826383
5.162e15   5521420198469 -1436027.5749  1.8291744043925245752  1.9021605826278
5.163e15   5522428851331 -1443439.9608  1.8291747951539250232  1.9021605826307
5.164e15   5523437455236 -1412694.5567  1.8291751858206854592  1.9021605826188
5.165e15   5524446055470 -1389075.0980  1.8291755764103794860  1.9021605826096
5.166e15   5525454713421 -1433967.3210  1.8291759669468059616  1.9021605826270
5.167e15   5526463272716 -1390995.9629  1.8291763573694516158  1.9021605826104
5.168e15   5527471848899 -1375702.7618  1.8291767477230794065  1.9021605826045
5.169e15   5528480434032 -1380147.4567  1.8291771380046457715  1.9021605826062
5.170e15   5529489009720 -1385932.7876  1.8291775282070605399  1.9021605826084
5.171e15   5530497559436 -1376529.4952  1.8291779183239608886  1.9021605826048
5.172e15   5531506129528 -1398283.3214  1.8291783083733061498  1.9021605826132
5.173e15   5532514633010 -1364206.0088  1.8291786983214877784  1.9021605826000
5.174e15   5533523173457 -1377870.3010  1.8291790882085862794  1.9021605826053
5.175e15   5534531671962 -1360366.9425  1.8291794780041263764  1.9021605825986
5.176e15   5535540134709 -1317877.6786  1.8291798677105319801  1.9021605825821
5.177e15   5536548659162 -1347864.2557  1.8291802573654950968  1.9021605825937
5.178e15   5537557146816 -1351819.4210  1.8291806469309829373  1.9021605825952
5.179e15   5538565620491 -1352560.9226  1.8291810364158447983  1.9021605825955
5.180e15   5539574070691 -1340590.5095  1.8291814258164450240  1.9021605825909
5.181e15   5540582514417 -1332906.9316  1.8291818151393782887  1.9021605825879
5.182e15   5541590945286 -1323124.9398  1.8291822043822123163  1.9021605825842
5.183e15   5542599430392 -1378336.2858  1.8291825935708696795  1.9021605826055
5.184e15   5543607858639 -1387442.7223  1.8291829826625057630  1.9021605826090
5.185e15   5544616253952 -1374367.0028  1.8291833716663892075  1.9021605826039
5.186e15   5545624658980 -1381755.8818  1.8291837605990012985  1.9021605826068
5.187e15   5546633050860 -1386744.1146  1.8291841494515549337  1.9021605826087
5.188e15   5547641370426 -1330163.4574  1.8291845382012683282  1.9021605825869
5.189e15   5548649759115 -1353448.6675  1.8291849269027021686  1.9021605825959
5.190e15   5549658113822 -1353492.5029  1.8291853155161365651  1.9021605825959
5.191e15   5550666475884 -1371629.7225  1.8291857040575354781  1.9021605826029
5.192e15   5551674832096 -1394653.0862  1.8291860925218389400  1.9021605826118
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6.598e15   6959685231330 -1067276.2892  1.8296661995107809321  1.9021605825246
6.599e15   6960680373712 -1110128.0423  1.8296665011376307529  1.9021605825376
6.600e15   6961675466780 -1111945.3837  1.8296668027038311843  1.9021605825381