Brilliant numbers

Brilliant numbers, as defined by Peter Wallrodt, are numbers with two prime factors of the same length (in decimal notation). These numbers are generally used for cryptographic purposes, and for testing the performance of prime factoring programs.

The challenge is to continue the table shown below. This table contains the least brilliant number with an even number of digits and the greatest brilliant number with an odd number of digits.

It is possible to define n-brilliant numbers as the product of n prime numbers of the same length.

Table of contents

Maximal and minimal 2-brilliant numbers

Minimal 3-brilliant numbers

Maximal 3-brilliant numbers

Minimal 4-brilliant numbers

Maximal 4-brilliant numbers

Maximal and minimal base-2 2-brilliant numbers

Density of brilliant numbers (by Richard Heylen)

Maximal and minimal 2-brilliant numbers

The next table contains the maximal and minimal numbers with given number of figures whose two prime factors have the same length when expressed in decimal notation.

Maximal and minimal 2-brilliant numbers
DigitsNumberPrime factorsDiscoverer
193 x
3
Jim Fougeron
2102 x
5
Dario Alpern
398923 x
43
Dario Alpern
4100317 x
59
Dario Alpern
599973257 x
389
Dario Alpern
6100013103 x
971
Dario Alpern
7107 - 1891583 x
6317
Dario Alpern
8107 + 432089 x
4787
Dario Alpern
9109 - 13725303 x
39521
Dario Alpern
10109 + 8126881 x
37201
Dario Alpern
111011 - 357231571 x
431833
Dario Alpern
121011 + 147281683 x
355009
Dario Alpern
131013 - 771380163 x
7245521
Dario Alpern
141013 + 731857929 x
5382337
Dario Alpern
151015 - 26118874013 x
52982903
Richard Heylen
161015 + 314902357 x
67103479
Dario Alpern
171017 - 297150425719 x
664779937
Richard Heylen
181017 + 831158267567 x
631841393
Dario Alpern
191019 - 5331809444697 x
5526557411
Richard Heylen
201019 + 491172005397 x
8532383917
Dario Alpern
211021 - 3532 1942092327 x
4 5574505161
Richard Heylen
221021 + 9871 0119972739 x
9 8814495433
Dario Alpern
231023 - 58719 8168943561 x
50 4619937933
Richard Heylen
241023 + 69125 3850475563 x
39 3932687257
Dario Alpern
251025 - 113270 1224339427 x
370 2024986981
Richard Heylen
261025 + 183163 4713216901 x
611 7280937483
Dario Alpern
271027 - 2811681 6039284931 x
5946 7035195149
Richard Heylen
281027 + 41531059 4044136579 x
9439 2659413907
Dario Alpern
291029 - 9327335 0346504617 x
36583 0888011371
Richard Heylen
301029 + 27926292 2036885471 x
38034 0884258249
Dario Alpern
311031 - 389185693 5855186693 x
538521 5634707327
Richard Heylen
321031 + 667117816 0270652879 x
848780 9552819573
Dario Alpern
331033 - 10832758682 6038690693 x
3624918 6426792769
Richard Heylen
341033 + 7092856715 9477900867 x
3500523 0421091927
Dario Alpern
351035 - 80726092950 4743206393 x
38324527 5762949601
Richard Heylen
361035 + 27731231547 3725119131 x
32018906 6546263567
Jim Fougeron
371037 - 707112407394 0762336703 x
889621192 8210070531
Richard Heylen
381037 + 1687291898446 0772347259 x
342584900 1386617493
Jim Fougeron
391039 - 10712216872853 9128208483 x
4510858609 8430581563
Richard Heylen
401039 + 9971828883498 3913311471 x
5467816844 9744921707
Jim Fougeron
411041 - 36991 2706241645 7894753317 x
7 8701478208 6546036553
Richard Heylen
421041 + 12072 2872845388 3196320979 x
4 3719965007 5309509133
Jim Fougeron
431043 - 2725 5149936898 7998869591 x
39 1926414779 6477703203
Richard Heylen
441043 + 9120 9416220529 8508378983 x
47 7517929351 3497904877
Jim Fougeron
451045 - 17153 8012743463 0196194863 x
650 1896712171 4506226241
Richard Heylen
461045 + 1411204 5492562244 7906868371 x
488 8798025755 5528444241
Jim Fougeron
471047 - 16772791 7031852805 8249521813 x
3582 0426944832 7527799271
Richard Heylen
481047 + 3932055 2256009243 0172482139 x
4865 6458908952 2586263787
Jim Fougeron
491049 - 47119436 4326829329 0666134453 x
51449 7704549507 1915161893
Richard Heylen
501049 + 95126471 4152186265 7404330791 x
37776 5975767080 0343659761
Jim Fougeron
511051 - 579291216 6130819063 0398411623 x
343387 0030343167 2610407227
Richard Heylen
521051 + 9793139533 7722542092 6914498491 x
716672 3753287142 5027689523
Jim Fougeron
531053 - 1671858626 3518991829 5875711699 x
5380317 5607522149 7635484067
Richard Heylen
541053 + 22171408301 9631082166 1891328463 x
7100749 8831637862 7095268359
Jim Fougeron
551055 - 26710492309 5218034300 4616980607 x
95307901 2701599053 8677972619
Richard Heylen
561055 + 622914602019 9335556486 0764167713 x
68483675 8578849654 5968810933
Jim Fougeron
571057 - 8349244705702 9837201547 5229439271 x
408654145 6970164126 1618151781
Jim Fougeron
581057 + 2317218276491 2198936605 3485370671 x
458134540 4680301502 5118660227
Jim Fougeron
591059 - 59731009891100 3114765934 0436549999 x
9902057753 4703898106 0013155973
Jim Fougeron
601059 + 2132139168817 8982927891 0570173437 x
4674712868 0684854707 6331371449
Peter Wallrodt
611061 - 691 0552178828 8211523215 4201320731 x
9 4767158159 6685321140 0570143201
Bouk de Water
621061 + 3991 9605820850 2188312577 4065415291 x
5 1005260511 1322566578 9755534589
Bouk de Water
631063 - 914 4992163448 9889675295 9323769763 x
68 9692446965 8108371268 4424134557
Bouk de Water
641063 + 1922 3505686289 8037778009 5228766733 x
44 7415909903 6844148191 4582608543
Jim Fougeron
651065 - 687158 9971766331 0591078685 6570509671 x
628 9419857483 0732390318 5090439303
Bouk de Water
661065 + 2317286 5952168341 3346502850 9515988163 x
348 9241764208 3274582598 6935358959
Jim Fougeron
671067 - 8091250 5110651609 5154757791 7550796961 x
7996 7305197038 8910336438 6584269431
Jim Fougeron
681067 + 6091678 0927835184 8046981335 7050416541 x
5959 1460604656 5611794502 9500096149
Jim Fougeron
691069 - 124110679 9049628454 5686213838 6056396401 x
93633 7920120938 1211489257 1488571159
Phil Carmody
701069 + 260724426 5973536430 8480998741 7412256391 x
40938 9807971291 5155256416 4844215577
Jim Fougeron
711071 - 7619231008 2702166394 9461270808 8690827091 x
432884 9348389996 0157713857 5168198191
Jim Fougeron
721071 + 11901162435 9550602349 2398975054 0378209139 x
615627 2480616606 0652634801 6196129359
Jim Fougeron
731073 - 82972279546 3843252090 7465709063 0001002097 x
4386837 6922543729 0923364184 1447115399
Richard Heylen
741073 + 105631093872 1328861786 2529696746 2523488383 x
9141836 3256179014 2787369950 1869702461
Jim Fougeron
751075 - 420911603482 2931805642 1367482412 2706652977 x
86181025 2072092157 5415636921 1739413183
Richard Heylen
761075 + 547311171602 0162596298 9233595908 9815716691 x
89512676 7445310892 2620197682 3759569403
Jim Fougeron
(2 May 2002)
771077 - 4851266574540 5469304796 5242036511 4117132441 x
375129597 1281810690 2056255934 4084771989
Richard Heylen
(10 May 2002)
781077 + 3185554311 4966205323 7177035161 1143673123 x
538925768 9214151307 3970633306 9119686561
Jim Fougeron
791079 - 56671729776145 3696959942 8207517694 6198769503 x
5781094869 8582916253 8759307497 1223276611
Richard Heylen
801079 + 59231333322076 5188990013 5038176080 7974795003 x
7500063320 1157802123 7780289418 0923803641
Richard Heylen
(16 May 2002)
811081 - 49892 8090949127 0712637360 1146061968 3106498869 x
3 5598654765 1499403487 7251052333 5003778519
Michael Bell
821081 + 175271 6185521526 1918373528 4137894011 0747187933 x
6 1783613112 6064522717 8709752383 9696139619
Richard Heylen
831083 - 1433319 9932327974 2988610690 4886345531 6009796077 x
50 0169237327 4167000023 1182095228 9282634671
Richard Heylen
(27 Jun 2002)
841083 + 856913 2087033835 8319180791 0364106785 8862243541 x
75 7076581220 5898287201 5989811643 3821618709
Richard Heylen
(27 Jun 2002)
851085 - 7007169 3200872034 1695875408 3626191779 6700786223 x
590 5973806868 0815745483 1212278304 4390802991
Richard Heylen
(8 Jul 2002)
861085 + 16701305 1275553674 2722241262 5346271954 4495568863 x
327 7317903313 6623331914 1570013682 9929609827
Richard Heylen
(27 Sep 2002)
871087 - 61131672 0583183841 0336460297 3032415415 4355596829 x
5980 6526423457 0267994286 2190355537 5545514603
Richard Heylen
(11 Oct 2002)
881087 + 119892554 8073574973 3420228780 0723585352 7655387529 x
3914 1894478478 0487593972 6201920459 1690357741
Reto Keiser
(25 Jul 2002)
891089 - 1454717182 5942024805 9924461238 7996462002 8962720089 x
58198 4296559615 5552330942 9349240261 7340927477
Richard Heylen
(24 Nov 2002)
901089 + 975712544 9117210655 5918816731 8368413723 3747818249 x
79713 5940240048 5345670349 5028670126 7149637493
Richard Heylen
(24 Nov 2002)
911091 - 35579297819 0131233156 6877668999 3983804239 6357404141 x
335774 3985223460 3802184405 8109031708 9334569081
David Cleaver
(5 Sep 2002)
921091 + 6489303042 6176188825 9047082534 8078377909 3519047793 x
329986 5899579960 5435086827 9346278922 2873063273
David Cleaver
(11 Nov 2002)
931093 - 136431308651 9402050960 4707836850 1221022392 5173521497 x
7641451 2467178771 4260836429 3152368823 0774568381
David Cleaver
(9 Dec 2002)
941093 + 34892941166 7593005444 7010239755 9827007826 4073129867 x
3400011 2262856379 6700171396 6956849633 0607787267
Jim Fougeron
(9 Dec 2002)
951095 - 368111109749 1776794870 9376300022 4384096761 6045495643 x
90011033 0131568068 2277608653 0971942745 8876830733
Richard Heylen
(30 Jan 2003)
961095 + 289913630093 2834042408 3940245923 3530210953 6892634161 x
73367069 4108588540 8146519774 5857415052 8292043459
Jim Fougeron
(12 Dec 2002)
971097 - 19163150730263 1098002948 3909591311 6970605940 8828799543 x
663436777 3056592241 5154945278 4246524717 6344374659
David Cleaver
(20 Mar 2003)
981097 + 12577124825954 2695921544 0349409041 9477333875 1647883781 x
801115445 7832187771 2536036138 9540856834 1998334317
Jim Fougeron
(3 Jan 2003)
991099 - 27833108480161 2134411476 3336994966 2780217266 0847517087 x
3217006215 6987845422 4355815533 0840713863 2546972991
David Cleaver
(13 Dec 2002)
1001099 + 40112315648505 7650669192 7684971633 4055819213 0875875893 x
4318444692 7518910400 6475667703 3452093865 5251359327
David Cleaver
(20 Dec 2002)
10110101 - 569931 1782454340 0831857004 2642818908 5916894417 2987388481 x
8 4871960555 6256173460 8988925131 0824028091 2733932447
David Cleaver
(4 Mar 2003)
10210101 + 32312 2701919653 5950614963 1764979892 5506904523 4236491673 x
4 4049138366 2192037893 8689413501 3226586083 9934720247
Richard Heylen
(13 Jul 2003)
10310103 - 241723 8868404401 3249161833 4280106561 3978194751 1337567461 x
41 8640549178 6561972002 2578515523 6366529229 1130223203
David Cleaver
(20 Jul 2003)
10410103 + 1738921 7885032974 4203343440 1041331778 9198161891 9402435883 x
45 8957637589 2693047888 0734959940 6651271487 1682387783
David Cleaver
(17 Aug 2003)
10510105 - 30179200 7284510838 9786097528 9736163621 0062604835 7754871711 x
498 1854812310 7522887110 5253775420 8478809696 2615971011
Richard Heylen
(14 Oct 2003)
10610105 + 4293128 4226997750 6686435684 4426458148 2168133598 8332440829 x
778 6785371678 8858218366 8785177552 7053995002 6013843817
Richard Heylen
(1 Aug 2003)
10710107 - 5692981 0893150893 5388275225 5462174328 6502046414 6509030899 x
3354 4784952879 7640933380 6165775647 0675149353 0526472669
Richard Heylen
(22 Oct 2003)
10810107 + 69692000 1790349395 8497496269 8258346499 2284082246 7385473869 x
4999 5524527143 3818753431 7433455459 0767522642 1199679901
Richard Heylen
(30 Oct 2003)
10910109 - 1061729392 9347145171 5027656817 5741564598 1896353292 9889785991 x
34021 7814149092 3281380298 5431644803 2470409630 1595136513
Richard Heylen
(30 Nov 2003)
11010109 + 1904127622 9153544139 4586840378 2275819821 0837314064 7210683969 x
36201 8268951545 3682000526 8087291994 7197343843 5837408289
David Cleaver
(16 Nov 2003)
11110111 - 11957141182 0397984271 8112107179 0859135842 7040257882 1772762747 x
708305 3916969546 1569685870 3508927026 7835105482 4192755569
Richard Heylen
(5 Apr 2004)
11210111 + 57339162769 0923050353 6141393381 7404912003 9148307877 6043167501 x
614367 2523073131 3013378040 0848273214 4910301727 0012524839
Sander Hoogendoorn
(15 Mar 2006)
11310113 - 912271354837 3134821631 0599104139 9619149091 5690067442 5643544869 x
7380959 9872166890 9879514364 9871648779 4623234653 1808934017
Sander Hoogendoorn
(20 Apr 2006)
11410113 + 293591299567 8224153860 6843279942 5018706158 2368298838 8592729389 x
7694865 8065524647 5123803852 8936909680 8671673449 0830614731
David Cleaver
(27 Mar 2007)
11510115 - 176911504120 0967794696 9526346072 3317602333 5486779937 1253396127 x
86925379 0457164835 6378342160 3635526344 0656363176 0612220553
David Cleaver
(1 Apr 2007)
11610115 + 1274116593409 1113277430 7356923305 1068375914 2603433470 9864681011 x
60264891 5175203383 9345332361 8479362122 0593165136 5703227431
Sander Hoogendoorn
(28 Apr 2006)
11710117 - 2733157262582 8389974101 2641225377 7830114415 2030142372 3971096511 x
635879165 8812967134 9443296973 9980973373 1062341237 4598098797
David Cleaver
(18 Dec 2007)
11810117 + 7341150501763 2843814481 3324771965 6235019180 5691855629 0873576027 x
664444042 4996512565 1345348476 3734158530 5429480908 1436607383
David Cleaver
(2 Oct 2008)
11910119 - 191071060245251 4498526274 2887581992 2502026799 9483770339 0958334023 x
9431780039 8778574370 9001406775 6729772644 6849617817 0211877691
David Cleaver
(16 Oct 2008)
12010119 + 104431733519903 6894877000 6118014625 4204082180 9720757108 9955436713 x
5768609854 8489607151 8016590923 2695970673 1224747267 5097328211
David Cleaver
(12 Apr 2010)
12110121 - 96331 3679005690 1172640709 6267890378 4274826594 0319877496 5236311639 x
7 3104728710 1046188617 2597258203 0918600850 5948605983 0493386553
David Cleaver
(13 Jan 2009)
12210121 + 74411 6050549491 5706436498 3042540776 2178545228 9715708181 8323775159 x
6 2303162924 4329337156 1630045207 4654938693 3502634075 8295841399
David Cleaver
(27 Mar 2010)
12310123 - 2645328 0732002176 0375200288 1036939514 1204456905 1168037124 4497994669 x
35 6211615437 0366033759 4063098262 2266482771 6307268511 0342932663
David Cleaver
(8 Apr 2010)
12410123 + 705123 4808021163 5120204597 3315154645 5783029514 2835223140 1178333959 x
42 5879829421 8557747573 1058591303 5641444527 8649483795 6270583389
David Cleaver
(23 Apr 2010)
12510125 - 16557166 5703358155 1519697547 2443830050 4891338492 9904334727 2954825491 x
600 3469916201 3395047367 9222482756 5444694530 9809096415 7383508673
David Cleaver
(27 Apr 2010)
12610125 + 4521132 2106277541 6234517159 3168355643 6144397989 2156046465 1676992659 x
756 3688464284 7278050326 7458846713 2288251246 4805056433 7149416019
David Cleaver
(23 Oct 2012)
12710127 - 29073004 6146242548 0114175254 8298266004 2163681505 6844271964 0543520887 x
3328 2138478841 2961926743 3103503463 1111424047 7541722530 5832697539
David Cleaver
(24 Oct 2012)
12810127 + 130091646 5124577863 8822072517 3031177402 2736202511 3306978968 2876014669 x
6073 4432665296 9607988725 3145345114 1404853786 2317634215 3392307861
David Cleaver
(27 Oct 2012)
12910129 - 5594910064 3878552488 4862571188 0039470176 5686284645 4651272528 7405454369 x
99360 2407203010 5010756776 9406969213 2116735641 9436356113 7033148179
David Cleaver
(11 Nov 2012)
13010129 + 12311664 4308485686 3136962963 7442720892 0783290888 4771675415 5425408347 x
85730 7152815529 1490396446 6215480820 8096809638 2724778190 7539975009
David Cleaver
(29 Apr 2010)
13110131 - 92459197427 0546102759 0950305557 3039977052 8045679849 8936939142 7647811271 x
506516 1925117181 2520125248 1400143562 4573028824 5245558499 6603306371
David Cleaver
(28 Dec 2013)
13210131 + 91041109528 5074437630 8645546283 5362783529 9757711413 9912969995 9296437089 x
913004 3158064994 6109965404 6315029952 2026636572 6210861515 1663407169
David Cleaver
(29 Sep 2014)
13310133 - 210993071414 3518267268 7378766911 9136020019 8591833148 9211667886 5902179259 x
3255829 0267975369 7845781898 1553165790 6568164764 8501887629 9731904239
David Cleaver
(25 Dec 2014)
13410133 + 24131494152 8028372203 6615875084 9004183379 1363514410 0174401041 0728805027 x
6692755 9089078283 2218859274 9508639420 2028395439 6040525253 6984044719
David Cleaver
(26 Dec 2014)
13510135 - 43120806007 8237600471 7798336653 1024716689 8008684206 2349960596 6016419309 x
48063040 6597280948 0574777434 5208478847 9150683268 9533275897 5323553141
David Cleaver
(30 Apr 2010)
13610135 + 1846321274932 0382385573 4511834899 4979437872 2737173093 9014203645 5070066967 x
47003675 4149271669 6885226327 0099126966 9770466918 1504850773 9406495289
David Cleaver
(1 Jan 2015)
13710137 - 23393115605278 7860292516 9798614498 3947697677 1775723061 5640765963 5859576889 x
865012402 9810727860 9987442135 1008178560 7659325917 8752159424 1686893463
David Cleaver
(9 Jan 2015)
13810137 + 17841163776634 7045626420 0924850769 4981115586 2764208395 3675453495 9460390339 x
610587707 9498575560 4475869051 0189671325 9966606123 4102523721 6625482619
David Cleaver
(14 Jan 2015)
13910139 - 327711458143858 1014743140 0544917914 0802992633 4618724206 1788926822 0382130409 x
6858033893 1853771284 4989669450 1312471774 6263440159 1184123680 9507604981
David Cleaver
(27 Jan 2015)
14010139 + 113891711572707 2776814018 8265343657 6961984319 6281408606 9774557920 1843022487 x
5842579726 5167679908 1469224446 6661242985 4045892595 9249543069 6082753547
David Cleaver
(14 May 2010)
14110141 - 456691 2113021658 3714674710 4649985070 6246065013 4548495912 8673352637 6480157581 x
8 2555784031 7149057049 4923559496 5960829095 3216319219 4268567086 5728631751
David Cleaver
(6 May 2015)
14210141 + 334533 1124911728 2728238224 2148600314 9122659266 4573224320 3951640140 2032184271 x
3 2128605174 2159193418 4519913284 1620333856 1063999436 1079097157 2320283843
David Cleaver
(17 Apr 2016)
14310143 - 2384113 5338143715 8648047268 9238134266 1682230201 4793595653 9350708690 7358098971 x
73 8889992535 6199085636 4480273022 6772032485 7281496718 7321971695 9251686029
David Cleaver
(17 Apr 2016)
14410143 + 8282113 0346521715 3628801868 7180980785 8006804650 2667829251 3682235182 3650537987 x
76 7185795861 6613897500 5269616708 2645623225 4543107978 5932201621 0304397783
David Cleaver
(17 Apr 2016)
14510145 - 40067214 3224364496 0829700335 7241579771 3808175407 4948412018 1981471914 4014601703 x
466 5867076567 6975118944 8254269956 6223786911 3542281868 7666926472 8008220411
David Cleaver
(11 Mar 2011)
14810147 + 191121 0754325987 9038482223 5833349894 6062147584 2503429633 0596159095 2868977237 x
8920 0063699717 0940867415 7068624024 3731188867 2010102299 2106066640 6453209287
Andreas Hglund
(29 Jul 2006)
15010149 + 2155319617 5124573517 0920349224 9342247090 8957491913 2952604732 0351896976 4669282857 x
50974 8624946200 8799446688 3520101722 0741864841 3016354748 0040592011 5982269129
Paul Leyland
(6 Jul 2006)
15510155 - 7030113748346 0682971758 2310178558 1604369932 9165454253 3072883571 3337978179 4162674443 x
72736021 8481797830 1985507698 8283737498 3036932333 9204705211 6566195061 2603395593
David Cleaver
(10 Sep 2012)

Density of brilliant numbers (by Richard Heylen)

Instead of deriving density results using other people's work I decided to think about the problem myself for a change and I'm quite pleased with the results.

With brilliant numbers we're trying to represent a number in the region of b^n (where n is odd) as the product of two primes where the larger of the two primes ranges from b^(n/2) to b^((n+1)/2). First we work out the average number of factors that a number b^n+-x has in this range. If we say that the factor in the range b^((n-1)/2) to b^ (n/2) is p and the larger factor is q. It's easy to see that they have the same number of digits base b. If we plot p against q then the number of ways is the number of lattice points on the curve p= (b^n+x)/q in the relevant region. We can estimate this by looking at the area between the curves p=b^n/q and p=(b^n+1)/q between the specified limits. I make this to be roughly ln b /2 which, for base 10 is about 1.15.

I initially found it surprising that a number would be representable as the product of two numbers in the relevant ranges in on average more than one way. After all, some of these numbers are prime or are the product of a tiny prime and a large prime so I looked a bit closer.

So that's 19 numbers representable in a total of 22 ways which works out to an average of 1.158 which is reasonable agreement. We then have to make sure that both p and q are prime. I can make a handwaving argument for the probabilities being independent based on the lack of usable modularity constraints. Anyway, when p and q are similar magnitude roughly one in n^2/4 ln^2 b will both be prime this goes down(up?) to one in (n^2-1) ln^2 b when they are at the other extremes of their ranges. The difference of (ln b)^2 can be usefully neglected when n is large enough to be interesting.

This means that we have ln b /2 ways of which only one in n^2/4 ln^2 b are acceptable. This cancels nicely to yield a density of 2- brilliants of one in n^2/2 ln b.

This gives excellent agreement with computed densities for instance the density of the first 1000 base 3 2-brilliants greater than 3^57 is one in 1765.118 and 57^2/2 ln 3 =1784.7

We can therefore expect that the density of 2-brilliants near 10^95 will be about one in 10390. The factor of ln b in the formula completely explains the density's base dependence left unexplained by the previous analysis.

From a handwavey point of view you can try and explain the base dependence by observing that 2^95+x is smaller than 10^95+x but then one has to mumble incoherently about being more likely to split into similar sized factors and I can't really justify it.

I have done a theoretical analysis of the density of 3-brilliants.
For numbers b^n±x the proportion of 3-brilliants is 1 in (n^3 * ln b)/9 for n=2 mod 3 and 1 in 4*(n^3* ln b)/9 for n=1 mod 3.

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Last updated on Apr 24th, 2016.