Enumeration of the prime quadruplets to 1e16

Thomas R. Nicely

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

Copyright © 2008 Thomas R. Nicely. All rights reserved. This document may be reproduced and distributed for educational and non-profit purposes.

Last updated 1000 GMT 9 September 2008.

This is a table of values of pi_4(x), the count of prime quadruplets (q, q+2, q+6, q+8) such that q <= x. The first three such quadruplets are (5, 7, 11, 13), (11, 13, 17, 19), and (101, 103, 107, 109). Also provided are the values of the related functions delta_4(x), S_4(x), and F_4(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.

Symbols are defined as follows; for further details, see the paper Enumeration to $1.6 \times 10^{15}$ of the prime quadruplets.

=============================================================================
    x           pi_4(x)  delta_4(x)         S_4(x)               F_4(x)
=============================================================================
1.0000e+01            1       10.29 0.51068931068931068931 0.964070321217938
2.0000e+01            2       10.13 0.78997658688061164841 0.995850537119856
3.0000e+01            2       10.52 0.78997658688061164841 0.930650926150766
4.0000e+01            2       10.78 0.78997658688061164841 0.900238875296448
5.0000e+01            2       10.98 0.78997658688061164841 0.882426432864690
6.0000e+01            2       11.14 0.78997658688061164841 0.870617832532254
7.0000e+01            2       11.28 0.78997658688061164841 0.862154610092485
8.0000e+01            2       11.40 0.78997658688061164841 0.855755297061471
9.0000e+01            2       11.51 0.78997658688061164841 0.850724023718443
1.0000e+02            2       11.60 0.78997658688061164841 0.846649213196690
2.0000e+02            4       10.28 0.84862463816744839238 0.885837792658963
3.0000e+02            4       10.73 0.84862463816744839238 0.878452459213742
4.0000e+02            4       11.09 0.84862463816744839238 0.874358881947354
5.0000e+02            4       11.39 0.84862463816744839238 0.871685161149193
6.0000e+02            4       11.65 0.84862463816744839238 0.869769052839529
7.0000e+02            4       11.89 0.84862463816744839238 0.868311278942790
8.0000e+02            4       12.10 0.84862463816744839238 0.867154911802740
9.0000e+02            5       11.30 0.85347319425313039516 0.871057486661900
1.0000e+03            5       11.49 0.85347319425313039516 0.870265083531968
2.0000e+03            7       10.97 0.85830014856295221497 0.870904337774445
3.0000e+03            8       11.08 0.86021861820386389916 0.871003212669616
4.0000e+03           10       10.02 0.86260190012786715074 0.872302766031642
5.0000e+03           10       10.85 0.86260190012786715074 0.871560103082273
6.0000e+03           11       10.61 0.86330923898742867245 0.871715934275308
7.0000e+03           11       11.31 0.86330923898742867245 0.871284429624808
8.0000e+03           11       11.96 0.86330923898742867245 0.870934200107401
9.0000e+03           11       12.58 0.86330923898742867245 0.870642099508084
1.0000e+04           12       12.17 0.86373319240018343749 0.870817270689693
2.0000e+04           19       10.09 0.86543869869358331799 0.871137006593710
3.0000e+04           22       11.05 0.86596668397742072590 0.871018723695818
4.0000e+04           24       12.52 0.86620756159806221122 0.870859205909501
5.0000e+04           25       14.67 0.86629891708557767456 0.870668654175279
6.0000e+04           27       15.60 0.86644910843780001277 0.870605186029452
7.0000e+04           30       15.35 0.86662968424609036084 0.870615852425834
8.0000e+04           33       14.97 0.86678702796906112160 0.870633421664927
9.0000e+04           36       14.47 0.86692977367980420769 0.870658251310591
1.0000e+05           38       14.88 0.86701100368413418480 0.870638051768363
2.0000e+05           52       21.64 0.86740403498179808375 0.870447599700749
3.0000e+05           70       21.09 0.86768899658409668347 0.870448343208639
4.0000e+05           87       19.74 0.86787972810538295799 0.870458542348862
5.0000e+05          103       18.21 0.86802351105101137206 0.870472992827201
6.0000e+05          117       17.82 0.86812515604747137518 0.870475311600818
7.0000e+05          133       14.76 0.86822366496408022150 0.870493989413471
8.0000e+05          141       19.16 0.86826662648062421767 0.870470695119222
9.0000e+05          156       16.11 0.86833754968946299463 0.870485299894500
1.0000e+06          166       17.68 0.86837953275349680687 0.870478518913352
2.0000e+06          295       -9.08 0.86875115958464441262 0.870563453719610
3.0000e+06          397      -22.72 0.86891935252277718154 0.870587818073754
4.0000e+06          467      -12.10 0.86900044121055585862 0.870575964861992
5.0000e+06          546      -15.68 0.86907139254206552425 0.870579524096043
6.0000e+06          606       -4.11 0.86911513490071583438 0.870571028240301
7.0000e+06          695      -24.54 0.86917004780686241981 0.870583640552685
8.0000e+06          772      -35.42 0.86921132018043422700 0.870589585161601
9.0000e+06          836      -35.36 0.86924143341928998178 0.870589507727509
1.0000e+07          899      -36.05 0.86926787696082905324 0.870589687487152
2.0000e+07         1468      -44.78 0.86942727158737936677 0.870592231357484
3.0000e+07         1952      -34.74 0.86950643077473118266 0.870591006938357
4.0000e+07         2404      -30.36 0.86955845087572910926 0.870590426522459
5.0000e+07         2847      -42.64 0.86959800487800295493 0.870591499323441
6.0000e+07         3258      -42.08 0.86962783621130663545 0.870591297288180
7.0000e+07         3647      -34.64 0.86965184685808291993 0.870590852496671
8.0000e+07         4034      -37.59 0.86967253048758811253 0.870591016496520
9.0000e+07         4402      -31.99 0.86968990025115242585 0.870590780038970
1.0000e+08         4768      -33.36 0.86970529363232326145 0.870590803418512
2.0000e+08         8097      -40.52 0.86979790703974411924 0.870590531328663
3.0000e+08        10973       57.76 0.86984469000401724839 0.870588938799399
4.0000e+08        13713       91.27 0.86987631749299986737 0.870588605980516
5.0000e+08        16331      108.54 0.86989969451376988659 0.870588441681838
6.0000e+08        18839      131.82 0.86991796256617200326 0.870588239303962
7.0000e+08        21275      144.52 0.86993299081504378794 0.870588162881789
8.0000e+08        23660      140.21 0.86994572656645063927 0.870588179086770
9.0000e+08        26082       41.35 0.86995713750309794580 0.870588642467898
1.0000e+09        28388        8.84 0.86996685642508722448 0.870588778250229
2.0000e+09        49262       58.50 0.87002514268475659175 0.870588611896844
3.0000e+09        68341      -60.15 0.87005609161903126524 0.870588731909245
4.0000e+09        85912      180.62 0.87007631769297149368 0.870588435740552
5.0000e+09       102950      157.13 0.87009153390081200390 0.870588453901904
6.0000e+09       119474       44.29 0.87010359599274363877 0.870588541649325
7.0000e+09       135206      242.07 0.87011330043036036008 0.870588420296803
8.0000e+09       150507      472.76 0.87012147809097871253 0.870588299913627
9.0000e+09       165764      409.58 0.87012866653505132186 0.870588328612067
1.0000e+10       180529      545.93 0.87013489117692849269 0.870588272187457
2.0000e+10       319107      234.71 0.87017357442665777759 0.870588357830011
3.0000e+10       445369      479.12 0.87019410213547726849 0.870588321011766
4.0000e+10       565240      164.20 0.87020790956094688591 0.870588355150816
5.0000e+10       679610      494.01 0.87021812647061302041 0.870588327324660
6.0000e+10       790510      600.85 0.87022621747464535866 0.870588319125475
7.0000e+10       898998      161.86 0.87023291100908785908 0.870588348208985
8.0000e+10      1004516      238.97 0.87023854866328073871 0.870588344419846
9.0000e+10      1107901      360.65 0.87024342012893015679 0.870588338456998
1.0000e+11      1209318      638.22 0.87024769554536542390 0.870588327409023
2.0000e+11      2161156    -1248.16 0.87027424475201135760 0.870588374628232
3.0000e+11      3037193    -1220.83 0.87028848298680229330 0.870588374666210
4.0000e+11      3870114    -2111.18 0.87029807841303991038 0.870588385745922
5.0000e+11      4671000    -1900.51 0.87030523156323761765 0.870588384091468
6.0000e+11      5449176    -2598.66 0.87031090927030425737 0.870588389435862
7.0000e+11      6207640    -2562.74 0.87031558730969668814 0.870588389182302
8.0000e+11      6949946    -2155.42 0.87031955316731335619 0.870588387155889
9.0000e+11      7680011    -2973.00 0.87032299338343396351 0.870588391074637
1.0000e+12      8398278    -3699.97 0.87032602081344096759 0.870588394083423
2.0000e+12     15133490     1440.62 0.87034480075276844714 0.870588379220184
3.0000e+12     21393554    -1222.10 0.87035497393944024876 0.870588384305203
4.0000e+12     27357696     2197.72 0.87036184329476933330 0.870588380061640
5.0000e+12     33122014      898.38 0.87036699142494094394 0.870588381340192
6.0000e+12     38728915      312.00 0.87037108207994458731 0.870588381843058
7.0000e+12     44210040     -752.07 0.87037446280276011544 0.870588382638302
8.0000e+12     49581777     2313.12 0.87037733251638009439 0.870588380957126
9.0000e+12     54864496     4438.70 0.87037982176320194686 0.870588379957074
1.0000e+13     60070590     4848.36 0.87038201608803389106 0.870588379770569
1.1000e+13     65205425     7277.05 0.87038397392095709017 0.870588378871596
1.2000e+13     70276639    11396.85 0.87038573906238776639 0.870588377417667
1.3000e+13     75297182    10249.67 0.87038734665375502820 0.870588377831101
1.4000e+13     80262909    12975.29 0.87038881875137318392 0.870588377031747
1.5000e+13     85186849    10767.85 0.87039017767208909650 0.870588377623505
1.6000e+13     90066180    10063.56 0.87039143735155746628 0.870588377807987
1.7000e+13     94908314     6576.22 0.87039261161736383736 0.870588378662067
1.8000e+13     99706681     9604.02 0.87039370870904388422 0.870588377954098
1.9000e+13    104475159     7669.47 0.87039474001564453933 0.870588378372780
2.0000e+13    109213708     2939.94 0.87039571227008331805 0.870588379346602
2.1000e+13    113916305     3335.84 0.87039663006257139528 0.870588379276147
2.2000e+13    118590700     2809.26 0.87039749990234176040 0.870588379381752
2.3000e+13    123237805     1983.08 0.87039832621522000555 0.870588379535802
2.4000e+13    127857201     2666.89 0.87039911263931239315 0.870588379429208
2.5000e+13    132451167     3847.07 0.87039986279763071287 0.870588379241878
2.6000e+13    137016789     9593.61 0.87040057907968089011 0.870588378342679
2.7000e+13    141567321     7712.38 0.87040126604397678344 0.870588378626140
2.8000e+13    146093569     8372.43 0.87040192448954803056 0.870588378530404
2.9000e+13    150598984     9022.55 0.87040255690162665872 0.870588378439014
3.0000e+13    155086540     7521.47 0.87040316545284183150 0.870588378645824
3.1000e+13    159557551     3327.95 0.87040375187468101409 0.870588379196286
3.2000e+13    164007487     1690.86 0.87040431699825054616 0.870588379402948
3.3000e+13    168441140    -1511.45 0.87040486272563396724 0.870588379795534
3.4000e+13    172856180    -3322.54 0.87040538994253252268 0.870588380015566
3.5000e+13    177253295    -3843.78 0.87040589979340977772 0.870588380076167
3.6000e+13    181634741    -4780.74 0.87040639351188932844 0.870588380179337
3.7000e+13    185999299    -4397.06 0.87040687185471992342 0.870588380138636
3.8000e+13    190350901    -6137.55 0.87040733605676416319 0.870588380324435
3.9000e+13    194684918    -4913.74 0.87040778637343920083 0.870588380197917
4.0000e+13    199006240    -5181.62 0.87040822400132247918 0.870588380224340
4.1000e+13    203311891    -3554.61 0.87040864927460369219 0.870588380063073
4.2000e+13    207605728    -3500.77 0.87040906315917093880 0.870588380056061
4.3000e+13    211885250    -2150.15 0.87040946595993705607 0.870588379930209
4.4000e+13    216156208    -4903.36 0.87040985871286467680 0.870588380185047
4.5000e+13    220412455    -5280.19 0.87041024131546585379 0.870588380219372
4.6000e+13    224651617     -589.44 0.87041061400551830660 0.870588379806539
4.7000e+13    228883691     -525.81 0.87041097806997729807 0.870588379800333
4.8000e+13    233102899      977.13 0.87041133338534486568 0.870588379672614
4.9000e+13    237313392       43.82 0.87041168065749757065 0.870588379750725
5.0000e+13    241512180      -72.46 0.87041201996742285091 0.870588379761043
5.1000e+13    245697652     2491.22 0.87041235150234680868 0.870588379558154
5.2000e+13    249872701     5083.12 0.87041267579056956857 0.870588379357926
5.3000e+13    254037464     7797.47 0.87041299311908820324 0.870588379153545
5.4000e+13    258198873     3924.07 0.87041330426152420850 0.870588379442001
5.5000e+13    262345548     5055.88 0.87041360861523044537 0.870588379359898
5.6000e+13    266478834    10052.47 0.87041390651840571056 0.870588378999384
5.7000e+13    270609323     8518.52 0.87041419895105460039 0.870588379108035
5.8000e+13    274729676     7982.28 0.87041448559131067273 0.870588379144207
5.9000e+13    278838729     9789.93 0.87041476655874741844 0.870588379019446
6.0000e+13    282940804     9795.01 0.87041504233619934326 0.870588379019498
6.1000e+13    287036384     7683.71 0.87041531312434332441 0.870588379158145
6.2000e+13    291121998     7090.26 0.87041557886219069828 0.870588379197007
6.3000e+13    295198132     7686.19 0.87041583974020183329 0.870588379158036
6.4000e+13    299261260    13149.62 0.87041609569080502227 0.870588378813451
6.5000e+13    303319414    15595.57 0.87041634736501162442 0.870588378661827
6.6000e+13    307372401    15359.15 0.87041659488165142565 0.870588378676836
6.7000e+13    311415150    17648.82 0.87041683806007932322 0.870588378539506
6.8000e+13    315454016    16242.64 0.87041707740585352337 0.870588378623476
6.9000e+13    319485809    14459.40 0.87041731284341813423 0.870588378727299
7.0000e+13    323510017    12935.90 0.87041754445682913342 0.870588378814804
7.1000e+13    327525071    13362.07 0.87041777226561260840 0.870588378790626
7.2000e+13    331535393    11433.18 0.87041799662320316424 0.870588378898335
7.3000e+13    335538773     9472.96 0.87041821750338705824 0.870588379006592
7.4000e+13    339532243    10559.79 0.87041843483896661712 0.870588378947550
7.5000e+13    343523277     7326.83 0.87041864912694043676 0.870588379121784
7.6000e+13    347505956     5797.16 0.87041886013354042435 0.870588379203095
7.7000e+13    351481625     4726.90 0.87041906801550380330 0.870588379259573
7.8000e+13    355452541     1957.33 0.87041927296895594406 0.870588379402270
7.9000e+13    359412768     3520.01 0.87041947476696639477 0.870588379322576
8.0000e+13    363366270     5543.90 0.87041967368850025912 0.870588379220987
8.1000e+13    367317843     3323.45 0.87041987004262404147 0.870588379331151
8.2000e+13    371261352     3081.68 0.87042006359202885735 0.870588379343237
8.3000e+13    375201389      312.32 0.87042025462681402004 0.870588379477568
8.4000e+13    379134141    -1088.15 0.87042044302491382287 0.870588379544919
8.5000e+13    383058932     -362.38 0.87042062881657873124 0.870588379510865
8.6000e+13    386980653    -2322.09 0.87042081229150054400 0.870588379602791
8.7000e+13    390896126    -3711.99 0.87042099335605911480 0.870588379667020
8.8000e+13    394809015    -8120.72 0.87042117223323322996 0.870588379868623
8.9000e+13    398711412    -7566.75 0.87042134861470750002 0.870588379843423
9.0000e+13    402608054    -6715.34 0.87042152276807729071 0.870588379805105
9.1000e+13    406499441    -5996.47 0.87042169476493798207 0.870588379773219
9.2000e+13    410387690    -7458.78 0.87042186474443717267 0.870588379836644
9.3000e+13    414267479    -5713.51 0.87042203252051939699 0.870588379760758
9.4000e+13    418145913    -7800.46 0.87042219844435882282 0.870588379849791
9.5000e+13    422018949    -9612.92 0.87042236238438647719 0.870588379926786
9.6000e+13    425884345    -8846.63 0.87042252428763655606 0.870588379894873
9.7000e+13    429743983    -7322.71 0.87042268427367772025 0.870588379831142
9.8000e+13    433600153    -7271.67 0.87042284247691706992 0.870588379828882
9.9000e+13    437450886    -6666.30 0.87042299885373141782 0.870588379804658
1.0000e+14    441296836    -6103.68 0.87042315346614045845 0.870588379781931
1.0100e+14    445137713    -5238.09 0.87042330633805992245 0.870588379747255
1.0200e+14    448970503    -1001.04 0.87042345738519324953 0.870588379580124
1.0300e+14    452803612    -1745.15 0.87042360697130110120 0.870588379609196
1.0400e+14    456631895    -2273.19 0.87042375492556026369 0.870588379629603
1.0500e+14    460455856    -3038.00 0.87042390129859616083 0.870588379658945
1.0600e+14    464276668    -5162.47 0.87042404616457486275 0.870588379739344
1.0700e+14    468092302    -6568.52 0.87042418947640197211 0.870588379792601
1.0800e+14    471904798    -9248.05 0.87042433133785194920 0.870588379892289
1.0900e+14    475711961   -10958.92 0.87042447169510860845 0.870588379955264
1.1000e+14    479511111    -8974.94 0.87042461047797686326 0.870588379882838
1.1100e+14    483306635    -7637.80 0.87042474787365866178 0.870588379834529
1.1200e+14    487101825   -10195.08 0.87042488402497305557 0.870588379926246
1.1300e+14    490888979    -8901.24 0.87042501868034842897 0.870588379880257
1.1400e+14    494673470    -9086.53 0.87042515205550551600 0.870588379886842
1.1500e+14    498449294    -4705.03 0.87042528396293892926 0.870588379733752
1.1600e+14    502224286    -3550.62 0.87042541469970130784 0.870588379693963
1.1700e+14    505996119    -3255.92 0.87042554420544292848 0.870588379683697
1.1800e+14    509760769      242.69 0.87042567236490260921 0.870588379564887
1.1900e+14    513521138     4082.12 0.87042579929832368292 0.870588379435573
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