Enumeration of the prime quadruplets to 1e16
Thomas R. Nicely
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.
- pi_4(x) = Number of prime quadruplets (q, q+2, q+6, q+8) such that
the first element q <= x .
- delta_4(x) = Li_4(x) - pi_4(x) .
A positive value indicates a deficit of quadruplets (compared to
the theoretical estimate); a negative value indicates a surplus of
quadruplets.
- Li_4(x) = integral(27/2*c_4/((ln(t))^4), t, 2, x) =
Hardy-Littlewood integral approximation for pi_4(x) . Although
this is the traditional formula, note that a slightly more
accurate (for small x) approximation is produced by the
asymptotically equivalent formula
Li_4*(x) = integral(27/2*c_4/((ln(t+6))^4), t, 2, x) .
- c_4 = Hardy-Littlewood quadruplets constant
= 0.30749 48787 58327 09312 33544 86071 07685 3....
The kth Hardy-Littlewood constant (k > 1) is defined as
c_k = prod((p^(k-1))*(p-k)/(p-1)^k, p; p prime, p > k) .
- S_4(x) = Sum of the reciprocals of all of the elements of the prime
quadruplets (q, q+2, q+6, q+8) such that q <= x . Note that if
x >= 11, the terms 1/11 and 1/13 are included twice.
- F_4(x) = First order extrapolation, from S_4(x),
of the Brun's constant B_4 of the quadruplets (the limit
of the sum of the reciprocals as x approaches +infinity):
B_4 \approx F_4(x) = S_4(x) + 18c_4/(ln(x))^3 .
- The values given for pi_4(x) are believed to be exact. The values
for delta_4(x), S_4(x), and F_4(x) are believed to be correct to all
digits shown, except for a possible (rounding) error of one ulp (one
unit in the last decimal place or least significant digit).
- Please inform me of any errors you find in these values.
=============================================================================
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
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