Thomas R. Nicely
Last updated 1000 GMT 18 January 2010.
This is an extended table of values of pi_4(x), for 1e16 <= x <= 2e16, of the counts of prime quadruplets (q, q+2, q+6, q+8) such that q <= x. Also provided are the values of the related functions delta_4(x), S_4(x), and F_4(x); see Enumeration of the prime quadruplets 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.
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x pi_4(x) delta_4(x) S_4(x) F_4(x)
========================================================================================
1.000000e+16 25379433651 7684.3412 0.87047769123404595410 0.870588379957320
1.000100e+16 25381686947 7727.7524 0.87047769213531932582 0.870588379957302
1.000200e+16 25383940000 7989.7009 0.87047769303640535945 0.870588379957197
1.000300e+16 25386194413 6867.1895 0.87047769393794515553 0.870588379957646
1.000400e+16 25388447374 7172.2209 0.87047769483881422888 0.870588379957524
1.000500e+16 25390699945 7842.7981 0.87047769573943732516 0.870588379957256
1.000600e+16 25392954681 6323.9236 0.87047769664083594323 0.870588379957863
1.000700e+16 25395209496 4701.6004 0.87047769754217604819 0.870588379958512
1.000800e+16 25397459779 7586.8311 0.87047769844161468855 0.870588379957359
1.000900e+16 25399715456 5053.6186 0.87047769934311920250 0.870588379958371
1.001000e+16 25401967315 6313.9657 0.87047770024300793004 0.870588379957867
1.001100e+16 25404223803 2920.8750 0.87047770114465636684 0.870588379959223
1.001200e+16 25406474710 5084.3494 0.87047770204398491230 0.870588379958359
1.001300e+16 25408727151 5689.3917 0.87047770294383650659 0.870588379958117
1.001400e+16 25410982403 3459.0045 0.87047770384472109929 0.870588379959008
1.001500e+16 25413234149 4710.1908 0.87047770474411535669 0.870588379958508
1.001600e+16 25415484101 7730.9532 0.87047770564270331700 0.870588379957302
1.001700e+16 25417733747 11033.2945 0.87047770654107943352 0.870588379955983
1.001800e+16 25419986114 11590.2176 0.87047770744045233409 0.870588379955761
1.001900e+16 25422240147 10456.7251 0.87047770834040062941 0.870588379956213
1.002000e+16 25424492496 10982.8198 0.87047770923958682646 0.870588379956003
1.002100e+16 25426745426 10903.5045 0.87047771013891523049 0.870588379956035
1.002200e+16 25428997109 12046.7820 0.87047771103765613724 0.870588379955579
1.002300e+16 25431250759 11198.6551 0.87047771193709241715 0.870588379955917
1.002400e+16 25433502733 12002.1264 0.87047771283577009001 0.870588379955596
1.002500e+16 25435753679 13809.1988 0.87047771373394793770 0.870588379954875
1.002600e+16 25438008732 11484.8750 0.87047771463367483164 0.870588379955803
1.002700e+16 25440261056 11865.1578 0.87047771553222326447 0.870588379955651
1.002800e+16 25442511437 14164.0499 0.87047771642990706221 0.870588379954734
1.002900e+16 25444765903 12353.5542 0.87047771732913068902 0.870588379955456
1.003000e+16 25447018341 12546.6733 0.87047771822745584679 0.870588379955379
1.003100e+16 25449269199 14295.4101 0.87047771912506135321 0.870588379954682
1.003200e+16 25451522116 13960.7672 0.87047772002339836476 0.870588379954815
1.003300e+16 25453773130 15504.7476 0.87047772092088715760 0.870588379954200
1.003400e+16 25456025544 15624.3538 0.87047772181884460401 0.870588379954152
1.003500e+16 25458279286 14391.5887 0.87047772271724194064 0.870588379954643
1.003600e+16 25460533339 12823.4551 0.87047772361567368922 0.870588379955268
1.003700e+16 25462785450 13172.9556 0.87047772451324197759 0.870588379955129
1.003800e+16 25465038778 12281.0931 0.87047772541120581090 0.870588379955484
1.003900e+16 25467291052 12418.8703 0.87047772630866023653 0.870588379955430
1.004000e+16 25469542506 13352.2899 0.87047772720569853605 0.870588379955058
1.004100e+16 25471793972 14249.3548 0.87047772810265230125 0.870588379954700
1.004200e+16 25474049066 11494.0676 0.87047772900096190032 0.870588379955798
1.004300e+16 25476300485 12389.4312 0.87047772989771827788 0.870588379955441
1.004400e+16 25478551409 13755.4482 0.87047773079418824674 0.870588379954897
1.004500e+16 25480803804 13626.1215 0.87047773169115475959 0.870588379954949
1.004600e+16 25483057516 12155.4537 0.87047773258855641854 0.870588379955534
1.004700e+16 25485308748 13140.4477 0.87047773348488129739 0.870588379955142
1.004800e+16 25487561013 13068.1062 0.87047773438152823963 0.870588379955171
1.004900e+16 25489813049 13200.4319 0.87047773527799475632 0.870588379955118
1.005000e+16 25492065154 13239.4276 0.87047773617439955507 0.870588379955103
1.005100e+16 25494314153 16360.0960 0.87047773706947900874 0.870588379953861
1.005200e+16 25496567428 15180.4399 0.87047773796617104285 0.870588379954330
1.005300e+16 25498819652 15027.4621 0.87047773886235562684 0.870588379954391
1.005400e+16 25501071567 15159.1652 0.87047773975832817148 0.870588379954339
1.005500e+16 25503323606 15142.5520 0.87047774065426093457 0.870588379954345
1.005600e+16 25505576214 14532.6254 0.87047774155033093654 0.870588379954588
1.005700e+16 25507829097 13623.3879 0.87047774244642127046 0.870588379954950
1.005800e+16 25510081290 13379.8424 0.87047774334214805407 0.870588379955046
1.005900e+16 25512334970 11624.9916 0.87047774423837709468 0.870588379955744
1.006000e+16 25514583908 14587.8383 0.87047774513263149044 0.870588379954566
1.006100e+16 25516833859 16513.3852 0.87047774602719978041 0.870588379953801
1.006200e+16 25519085985 16239.6350 0.87047774692254381388 0.870588379953909
1.006300e+16 25521338417 15635.5905 0.87047774781792056356 0.870588379954150
1.006400e+16 25523589613 16243.2544 0.87047774871271697865 0.870588379953908
1.006500e+16 25525840605 17030.6295 0.87047774960734344966 0.870588379953595
1.006600e+16 25528095047 14343.7185 0.87047775050325205075 0.870588379954663
1.006700e+16 25530347175 13946.5241 0.87047775139815217427 0.870588379954821
1.006800e+16 25532599126 13702.0491 0.87047775229289305419 0.870588379954918
1.006900e+16 25534850529 13981.2963 0.87047775318732738682 0.870588379954807
1.007000e+16 25537099991 16177.2683 0.87047775408090185090 0.870588379953935
1.007100e+16 25539352432 15369.9679 0.87047775497557085271 0.870588379954255
1.007200e+16 25541604755 14656.3978 0.87047775587010413920 0.870588379954539
1.007300e+16 25543855687 15309.5609 0.87047775676399625035 0.870588379954279
1.007400e+16 25546107846 14711.4597 0.87047775765828680534 0.870588379954517
1.007500e+16 25548360421 13673.0971 0.87047775855265378130 0.870588379954929
1.007600e+16 25550613685 11921.4757 0.87047775944720551698 0.870588379955624
1.007700e+16 25552865340 11754.5984 0.87047776034102975346 0.870588379955691
1.007800e+16 25555118273 10285.4678 0.87047776123527255843 0.870588379956274
1.007900e+16 25557369057 10941.0867 0.87047776212857374257 0.870588379956014
1.008000e+16 25559620275 11138.4578 0.87047776302195853661 0.870588379955935
1.008100e+16 25561871584 11220.5839 0.87047776391529082705 0.870588379955903
1.008200e+16 25564125096 9075.4676 0.87047776480940856251 0.870588379956754
1.008300e+16 25566374752 10762.1118 0.87047776570190783196 0.870588379956085
1.008400e+16 25568627805 9027.5191 0.87047776659566614562 0.870588379956773
1.008500e+16 25570881750 6376.6922 0.87047776748968963131 0.870588379957824
1.008600e+16 25573131859 7537.6340 0.87047776838210309842 0.870588379957364
1.008700e+16 25575384607 6035.3471 0.87047776927547465466 0.870588379957960
1.008800e+16 25577635431 6432.8342 0.87047777016799466004 0.870588379957802
1.008900e+16 25579885308 7753.0981 0.87047777106005075823 0.870588379957278
1.009000e+16 25582139546 4688.1415 0.87047777195374735807 0.870588379958494
1.009100e+16 25584391528 3854.9672 0.87047777284646109016 0.870588379958824
1.009200e+16 25586642332 4175.5778 0.87047777373861940629 0.870588379958697
1.009300e+16 25588893813 3794.9761 0.87047777463095764127 0.870588379958847
1.009400e+16 25591144682 4002.1648 0.87047777552296496921 0.870588379958765
1.009500e+16 25593394783 4953.1466 0.87047777641457963229 0.870588379958389
1.009600e+16 25595643845 6918.9243 0.87047777730569428348 0.870588379957610
1.009700e+16 25597895037 6730.5006 0.87047777819756454336 0.870588379957684
1.009800e+16 25600147000 5746.8781 0.87047777908965188608 0.870588379958074
1.009900e+16 25602397150 6552.0597 0.87047777998093278875 0.870588379957755
1.010000e+16 25604648908 5725.0480 0.87047778087276229072 0.870588379958083
1.010100e+16 25606898522 7017.8458 0.87047778176365444554 0.870588379957571
1.010200e+16 25609149310 7112.4558 0.87047778265492325968 0.870588379957533
1.010300e+16 25611401496 5784.8807 0.87047778354665739754 0.870588379958059
1.010400e+16 25613650349 7766.1232 0.87047778443698372746 0.870588379957274
1.010500e+16 25615898284 10641.1860 0.87047778532685855452 0.870588379956136
1.010600e+16 25618149305 10406.0719 0.87047778621786679687 0.870588379956229
1.010700e+16 25620399366 11106.7836 0.87047778710840694402 0.870588379955952
1.010800e+16 25622651453 9757.3237 0.87047778799966076708 0.870588379956486
1.010900e+16 25624903277 8646.6951 0.87047778889072237653 0.870588379956926
1.011000e+16 25627153166 9446.9004 0.87047778978093020995 0.870588379956609
1.011100e+16 25629402468 10809.9423 0.87047779067081776575 0.870588379956070
1.011200e+16 25631653626 10292.8235 0.87047779156135150070 0.870588379956274
1.011300e+16 25633903973 10562.5469 0.87047779245147639654 0.870588379956168
1.011400e+16 25636155937 9191.1150 0.87047779334215283525 0.870588379956710
1.011500e+16 25638407536 8160.5306 0.87047779423259687981 0.870588379957118
1.011600e+16 25640656567 9673.7963 0.87047779512193741048 0.870588379956519
1.011700e+16 25642908468 8292.9150 0.87047779601232481243 0.870588379957065
1.011800e+16 25645156071 11185.8894 0.87047779690092497044 0.870588379955922
1.011900e+16 25647406918 10810.7220 0.87047779779071971068 0.870588379956070
1.012000e+16 25649658209 9967.4157 0.87047779868060197969 0.870588379956403
1.012100e+16 25651910405 8194.9731 0.87047779957075405161 0.870588379957104
1.012200e+16 25654162509 6490.3970 0.87047780046078179490 0.870588379957777
1.012300e+16 25656412507 6867.6901 0.87047780134988944358 0.870588379957628
1.012400e+16 25658664055 5670.8550 0.87047780223952169169 0.870588379958101
1.012500e+16 25660916338 3714.8945 0.87047780312935643081 0.870588379958874
1.012600e+16 25663166396 3959.8113 0.87047780401822433078 0.870588379958777
1.012700e+16 25665415851 4783.6081 0.87047780490676629668 0.870588379958452
1.012800e+16 25667665906 4983.2876 0.87047780579545746259 0.870588379958373
1.012900e+16 25669915603 5516.8525 0.87047780668391951713 0.870588379958162
1.013000e+16 25672164667 6659.3055 0.87047780757204393103 0.870588379957711
1.013100e+16 25674415137 6371.6493 0.87047780846063583157 0.870588379957825
1.013200e+16 25676663540 8126.8866 0.87047780934832393259 0.870588379957132
1.013300e+16 25678912179 9622.0201 0.87047781023601760602 0.870588379956541
1.013400e+16 25681160388 11523.0525 0.87047781112345396780 0.870588379955791
1.013500e+16 25683410281 11715.9866 0.87047781201146736915 0.870588379955715
1.013600e+16 25685660265 11793.8249 0.87047781289942907894 0.870588379955684
1.013700e+16 25687910920 11176.5703 0.87047781378756795870 0.870588379955928
1.013800e+16 25690160292 11818.2254 0.87047781467511303184 0.870588379955675
1.013900e+16 25692410442 11657.7929 0.87047781556287747352 0.870588379955738
1.014000e+16 25694660847 11218.2755 0.87047781645065501768 0.870588379955911
1.014100e+16 25696910555 11451.6759 0.87047781733807003126 0.870588379955819
1.014200e+16 25699159561 12362.9968 0.87047781822512065530 0.870588379955460
1.014300e+16 25701411175 10642.2409 0.87047781911311237565 0.870588379956138
1.014400e+16 25703659601 12085.4109 0.87047781999975940396 0.870588379955569
1.014500e+16 25705912051 9480.5095 0.87047782088790568402 0.870588379956596
1.014600e+16 25708162700 8652.5394 0.87047782177525434818 0.870588379956923
1.014700e+16 25710409495 11654.5032 0.87047782266099619346 0.870588379955739
1.014800e+16 25712661075 9847.4038 0.87047782354853697279 0.870588379956452
1.014900e+16 25714913177 7494.2437 0.87047782443619603722 0.870588379957379
1.015000e+16 25717163458 6938.0256 0.87047782532304996380 0.870588379957598
1.015100e+16 25719412750 7346.7523 0.87047782620942676599 0.870588379957437
1.015200e+16 25721660612 9161.4265 0.87047782709515281312 0.870588379956722
1.015300e+16 25723910370 9056.0508 0.87047782798153862117 0.870588379956764
1.015400e+16 25726160545 8509.6279 0.87047782886800142863 0.870588379956979
1.015500e+16 25728411578 7081.1605 0.87047782975471490176 0.870588379957542
1.015600e+16 25730660441 7798.6513 0.87047783064048634366 0.870588379957259
1.015700e+16 25732907175 10621.1030 0.87047783152533210840 0.870588379956147
1.015800e+16 25735156796 10532.5183 0.87047783241122765797 0.870588379956182
1.015900e+16 25737406065 10771.8999 0.87047783329689741709 0.870588379956088
1.016000e+16 25739657019 9302.2503 0.87047783418314339432 0.870588379956667
1.016100e+16 25741906449 9332.5725 0.87047783506870220939 0.870588379956655
1.016200e+16 25744156362 8855.8689 0.87047783595436396868 0.870588379956843
1.016300e+16 25746405160 9470.1424 0.87047783683949970839 0.870588379956601
1.016400e+16 25748655707 8311.3956 0.87047783772523667868 0.870588379957057
1.016500e+16 25750906747 6635.6311 0.87047783861108056508 0.870588379957716
1.016600e+16 25753155929 6793.8517 0.87047783949610615070 0.870588379957654
1.016700e+16 25755407507 4532.0600 0.87047784038198745235 0.870588379958544
1.016800e+16 25757658711 2620.2588 0.87047784126763447204 0.870588379959296
1.016900e+16 25759906358 4241.4506 0.87047784215179517625 0.870588379958658
1.017000e+16 25762155546 4297.6383 0.87047784303647506107 0.870588379958636
1.017100e+16 25764404083 4980.8244 0.87047784392081193626 0.870588379958367
1.017200e+16 25766653413 4847.0116 0.87047784480537370461 0.870588379958420
1.017300e+16 25768902904 4528.2027 0.87047784568991182234 0.870588379958545
1.017400e+16 25771152282 4298.4003 0.87047784657431857562 0.870588379958636
1.017500e+16 25773400545 5159.6071 0.87047784745820005272 0.870588379958297
1.017600e+16 25775651530 3274.8257 0.87047784834306469042 0.870588379959038
1.017700e+16 25777900362 3519.0589 0.87047784922699610135 0.870588379958942
1.017800e+16 25780150408 2525.3093 0.87047785011131778972 0.870588379959333
1.017900e+16 25782397555 4406.5795 0.87047785099441332686 0.870588379958593
1.018000e+16 25784647038 3927.8724 0.87047785187834003314 0.870588379958782
1.018100e+16 25786893238 6708.1904 0.87047785276089002487 0.870588379957689
1.018200e+16 25789141310 7592.5364 0.87047785364408877815 0.870588379957342
1.018300e+16 25791389349 8485.9130 0.87047785452718779221 0.870588379956991
1.018400e+16 25793637682 9061.3229 0.87047785541031559943 0.870588379956765
1.018500e+16 25795889524 6103.7686 0.87047785629473486956 0.870588379957926
1.018600e+16 25798137432 7056.2530 0.87047785717752237233 0.870588379957552
1.018700e+16 25800386747 6577.7787 0.87047785806077570392 0.870588379957740
1.018800e+16 25802634618 7519.3483 0.87047785894337535455 0.870588379957370
1.018900e+16 25804884535 6390.9645 0.87047785982669163110 0.870588379957813
1.019000e+16 25807134786 4904.6300 0.87047786071005238137 0.870588379958397
1.019100e+16 25809384586 3845.3475 0.87047786159314936784 0.870588379958813
1.019200e+16 25811633860 3288.1196 0.87047786247595326447 0.870588379959031
1.019300e+16 25813883947 1893.9490 0.87047786335898961619 0.870588379959579
1.019400e+16 25816131933 2576.8384 0.87047786424111485208 0.870588379959311
1.019500e+16 25818379461 3693.7904 0.87047786512297389606 0.870588379958872
1.019600e+16 25820628170 3605.8077 0.87047786600520977171 0.870588379958907
1.019700e+16 25822876703 3669.8929 0.87047786688729010041 0.870588379958882
1.019800e+16 25825123558 5388.0488 0.87047786776862575836 0.870588379958208
1.019900e+16 25827372828 4667.2779 0.87047786865082215862 0.870588379958490
1.020000e+16 25829620288 5732.5830 0.87047786953222222557 0.870588379958073
1.020100e+16 25831870832 3689.9668 0.87047787041474519211 0.870588379958874
1.020200e+16 25834118507 4492.4318 0.87047787129605676462 0.870588379958559
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