Tables relating to Carmichael numbers

Lists of Carmichael numbers

Statistics on Carmichael numbers

Distribution of Carmichael numbers. The number of Carmichael numbers less than M with k prime factors
M
k 103 104 105 106 107 108 10910101011
31712234784172335590
400419551443146191179
500013271464921336
60000001499459
70000000241
Total17164310525564615473605

M
k1012101310141015101610171018101910201021
3100018583284608310816195393558665309120625224763
421023639604299381620225758406856334398253151566
531567082149382928255012100707178063306310514381846627
61714527014401369078669619430641466084956416817443230120
7262134053591921060150172234460553115916727747026363475
87896553622163486363522399772040621480176015901
9012717014368835449931963917629632714473
10000023340305820738114232547528
1100000149576580442764
120000000256983
Total8241192794470610521224668358535514016443381806822077720138200

M
k102210231024102510261027102810291030
1210018-
13551277-
1400161159921568
150002871790
16000000511642
170000000142

The smallest Carmichael numbers with k prime factors:
knumberfactors
3 561 3 11 17
4 41041 7 11 13 41
5 825265 5 7 17 19 73
6 321197185 5 19 23 29 37 137
7 5394826801 7 13 17 23 31 67 73
8 232250619601 7 11 13 17 31 37 73 163
9 9746347772161 7 11 13 17 19 31 37 41 641
10 1436697831295441 11 13 19 29 31 37 41 43 71 127
11 60977817398996785 5 7 17 19 23 37 53 73 79 89 233
12 7156857700403137441 11 13 17 19 29 37 41 43 61 97 109 127
13 1791562810662585767521 11 13 17 19 31 37 43 71 73 97 109 113 127
14 87674969936234821377601 7 13 17 19 23 31 37 41 61 67 89 163 193 241
15 6553130926752006031481761 11 13 17 19 29 31 41 43 61 71 73 109 113 127 181
16 1590231231043178376951698401 17 19 23 29 31 37 41 43 61 67 71 73 79 97 113 199
17 35237869211718889547310642241 13 17 19 23 29 31 37 41 43 61 67 71 73 97 113 127 211
18 32809426840359564991177172754241 13 17 19 23 29 31 37 41 43 61 67 71 73 97 127 199 281 397
19 2810864562635368426005268142616001 13 17 19 23 29 31 37 41 43 61 67 71 73 109 113 127 151 281 353
20 349407515342287435050603204719587201 11 13 17 19 29 31 37 41 43 61 71 73 97 101 109 113 151 181 193 641
21 125861887849639969847638681038680787361 13 17 19 23 29 31 37 41 43 61 67 71 73 89 97 113 181 211 241 331 353
22 12758106140074522771498516740500829830401 13 17 19 23 29 31 37 41 43 61 67 71 73 89 97 101 113 127 181 193 211 1153
23 2333379336546216408131111533710540349903201 11 13 17 19 29 31 37 41 43 47 61 71 73 101 109 113 127 139 163 211 337 421 541
24 294571791067375389885907239089503408618560001 11 13 17 19 31 37 41 43 47 59 61 71 73 97 101 109 113 127 151 181 211 251 257 631
25 130912961974316767723865201454187955056178415601 11 13 17 19 29 31 37 41 43 47 61 71 73 101 109 113 127 139 151 181 211 241 281 541 701
26 13513093081489380840188651246675032067011140079201 11 17 19 29 31 37 41 43 47 53 61 71 73 79 97 101 109 127 151 157 163 179 271 281 337 433
27 7482895937713262392883306949172917048928068129206401 13 17 19 23 29 31 37 41 43 61 67 71 73 89 97 101 109 113 127 151 199 211 271 331 337 397 601
28 1320340354477450170682291329830138947225695029536281601 13 17 19 23 29 31 37 41 43 61 67 73 89 97 101 109 113 127 151 181 193 199 211 241 257 281 331 449
29 379382381447399527322618466130154668512652910714224209601 17 19 23 29 31 37 41 43 53 61 67 71 73 79 89 97 101 113 127 131 151 157 181 193 271 281 401 433 547
30 70416887142533176417390411931483993124120785701395296424001 11 17 19 29 31 37 41 43 47 53 61 71 73 79 97 101 109 113 127 131 139 157 163 167 241 313 449 461 487 599
31 2884167509593581480205474627684686008624483147814647841436801 17 19 23 29 31 37 41 43 53 61 67 71 73 79 89 97 101 109 113 127 131 151 157 163 181 199 211 241 271 353 617
32 4754868377601046732119933839981363081972014948522510826417784001 17 19 23 29 31 37 41 43 53 61 67 71 73 79 89 97 101 109 127 131 151 157 181 193 199 211 241 271 313 331 353 937
33 1334733877147062382486934807105197899496002201113849920496510541601 17 19 23 29 31 37 41 43 53 61 67 71 73 79 89 97 101 109 113 127 131 151 157 163 181 211 271 313 331 337 379 401 911
34 260849323075371835669784094383812120359260783810157225730623388382401 17 19 29 31 37 41 43 47 53 61 67 71 73 79 89 97 101 109 113 127 131 139 151 157 163 181 193 211 277 281 353 421 487 829

The ratio C(10n)/C(10n-1)
nC(10n)C(10n-1)ratio
4717.000
51672.286
643162.688
7105432.441
82551052.429
96462552.533
1015476462.395
11360515472.330
12824136052.286
131927982412.339
1444706192792.319
15105212447062.353
162466831052122.345
175853552466832.373
1814016445853552.394
19338180614016442.413
20822077733818062.431
212013820082207772.450

The function k(x) of Pomerance/Selfridge/Wagstaff
nk(x)
3 2.93319
4 2.19547
5 2.07632
6 1.97946
7 1.93388
8 1.90495
9 1.87989
10 1.86870
11 1.86421
12 1.86377
13 1.86240
14 1.86293
15 1.86301
16 1.86406
17 1.86472
18 1.86522
19 1.86565
20 1.86598
21 1.86619

Distribution into residue classes
mc25.10910111012101310141015
50203312627133027735814
1165227856575157553746790167
28215432770214843048
310217234472514633059
412418236876715193124
7040163413342774589112691
1109618854613114472800169131
210518643296721094599
3152232496105521784707
412921145098521224592
5138222454103322244777
6142235462101821814715
11033554713243006703216563
16401131277067861654840891
2139217473106823615338
3142220457104523485319
4104187442102623175261
5152243466106623705316
6116198440106124005384
7122195458102322235165
8129222475110724505449
9131218465104222855179
10153227471104923725347
1212071346279691876143760103428
3001225
5203264124228448
747751472895471027
9253660103165294
110000410

Largest first and largest last prime factors
Largest first prime and largest last prime up to 1015
651693055693681 72931 87517 102103
949803513811921 17 31 191 433 21792241
Largest first prime and largest last prime up to 1016
9585921133193329 174763 199729 274627
9463098235353841 13 31 541 631 68786257
Largest first prime and largest last prime up to 1017
90256390764228001 380251 410671 577981
99816335969903281 17 31 379 2237 223401361

Number of times a prime appears as least factor
p25.1091011101210131014101510161017
32536611051672995651025
52023096241325276557971217525481
7364579121825575461118742591557459
11263428107125095979143973389380745
13237431105824625699135143202576256
171172064961318324481142020650170
191522445321401335881412002049413
23377820753513603317819520803
2955103284729182246591157729149
31101168390876211651531257530667
37609521955114013418859421382
41356817141410922736678817275
4335651684039432308552013636
471416368119545911352854
5319305514736397323275842
592411431002726181542
6134581483648511978472211278
678185012331781519504843
7115256616138997924806178
73142868175406101525086277
79410176617546711632873
8311483979175457
891016235514840910032523
9710205010626160614133445

Number of times a prime appears as a factor
p25.10910111012101310141015
3253661105167299
5203312627133027735814
740163413342774589112691
1133554713243006703216563
1348380717843998904520758
1729348911822817664016019
1937260813553345779718638
23113207507128231357716
291943368322094515812721
3133557113203086727017382
3732053512702926682616220
4122739010012418589614344
431842967721920466311594
47538019949212232873
539216035181320415143
592641922626441611
6126945310752542604714429
67110178407106325406306
71104194521132033518546
731983488492145492511929
796410724768617284318
83142456137340838
896813132078819514981
97123193495127731237594

The strong Fibonacci pseudoprimes up to 1017
typeNfactors
LC-2829530326392129 31 67 271 331 5237
uLC-44337288862944117 31 41 43 89 97 167 331
sLC-58292008086312141 53 79 103 239 271 509
LC-89422110577800117 23 29 31 79 89 181 1999
LC-201374533760400117 37 41 131 251 571 4159
uLC-3967114933349568117 37 41 71 79 97 113 131 191

The index of a Carmichael number is i(n) = (n-1)/lambda(n).
Carmichael numbers known to have index up to 100 (complete for n up to 1021).
iNfactors
566017 23 41
75613 11 17
185546217717 23 83 1709
18888525144111 47 1109 15497
184201833384111 47 1049 77477
21105855 29 73
2224655 17 29
2311055 13 17
25119210013 29 263 521
31627453 5 47 89
371197201743 433 643
376790203143 271 5827
3933415319 43 409
43526337 73 103
44158417 31 73
4589117 19 67
4728217 13 31
4817297 13 19
49120836123747866953 653 26479 1318579
50419993280129 499 503 577
5220695584117 71 277 619
53127132584117 31 179 13477
54416986768913 29 383 28879
5527179460113 19 743 1481
6068400017 17 229 251
6119628045655 103 149 25579
6741004141 73 137
671198592499508390129 101 1427 16349 175403
7016240117 41 233
76475271776111 17 107 173 1373
81355750758095055 197 223 353 458807
8214964059337403455 47 317 40253 499027
831421599589241855 37 107 58379 123017
831586647618998855 37 107 53987 148469
832043703701402855 37 107 48677 212099
83248319081051242055 29 719 3023 78790717
90377811804057370200111 47 1051 67967 102302009
9252017898296111 29 131 607 20507
94127828490655 7 269 317 4283
95895691160111 17 127 131 2879
975472940991761199 241 863 132233
97721574219707441167 241 5039 3557977
979729822470631481127 409 110681 1692407
9783565865434172201103 1993 9551 42622169
9943825396587033743 139 49409 1484009

The Lehmer index of N is l(N) = (N-1)/phi(N).
Carmichael numbers with Lehmer index >= 2.
l(N)Nfactors
2.00162663735634039256198347138529719419600653 5 23 89 113 1409 788129
2.0038836128653465242496046406555105494434653 5 23 53 389 2663 34607
2.008372508329582505406681230134626273330989453 5 23 53 197 8009 466649
2.010009765625000000000000000267082533189681453 5 17 113 57839 16025297
2.0306736680327868852459016392690409923995653 5 23 29 4637 5799149
2.0320734374624290660767529091583536589323053 5 17 89 149 563 83177
2.03613545116795955722130218718176713599792453 5 23 29 359 11027 45893
2.071377818273530507776492047160571907822347853 5 17 29 269 6089 1325663
2.072936866685888594678744261751316424159741453 5 23 29 53 617 9857 23297
2.0812877308238636363636363638817155044507053 5 17 47 89 113 503 14543
2.083805744941863020924916476314541438588201453 5 17 23 2129 39293 64109
2.08893601543589668667430777561286139216727053 5 17 23 353 7673 385793
2.098407897763451927081664691123015767524089453 5 23 29 53 113 197 1042133
2.11432237861570247933884297518866163736653 5 17 23 83 353 10979
2.11493309048018742621015348231932315389891853 5 17 23 113 167 2927 9857
2.139359882439079747328472147119478165235869453 5 17 23 89 113 233 617 1409


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Updated 24/03/2008 by Richard Pinch