Tables relating to Carmichael numbers

Lists of Carmichael numbers

• carmichael-16.gz All Carmichael numbers up to 10¹⁶ in GZIP format
• carmichael17.gz All Carmichael numbers between 10¹⁶ and 10¹⁷ in GZIP format
• carmichael18.gz All Carmichael numbers between 10¹⁷ and 10¹⁸ in GZIP format
• car3-18.gz All Carmichael numbers with 3 prime factors up to 10¹⁸ in GZIP format

Statistics on Carmichael numbers

• Distribution of Carmichael numbers.
• The smallest Carmichael numbers with k prime factors for k up to 34
• The ratio C(10ⁿ)/C(10^n-1) with plot of ratio (PS)
• The function k(x) of Pomerance/Selfridge/Wagstaff with plot of k(x) (PS)
• Distribution into residue classes
• Largest first and largest last prime factors
• Number of times a prime appears as least factor
• Number of times a prime appears as a factor
• The strong Fibonacci pseudoprimes up to 10¹⁷
• Known to have index up to 100 (complete for n up to 10¹⁷)
• Lehmer index >= 2

Distribution of Carmichael numbers. The number of Carmichael numbers less than M with k prime factors

	M
k	103	104	105	106	107	108	109	1010	1011
3	1	7	12	23	47	84	172	335	590
4	0	0	4	19	55	144	314	619	1179
5	0	0	0	1	3	27	146	492	1336
6	0	0	0	0	0	0	14	99	459
7	0	0	0	0	0	0	0	2	41
Total	1	7	16	43	105	255	646	1547	3605

	M
k	1012	1013	1014	1015	1016	1017	1018	1019	1020	1021
3	1000	1858	3284	6083	10816	19539	35586	65309	120625	224763
4	2102	3639	6042	9938	16202	25758	40685	63343	98253	151566
5	3156	7082	14938	29282	55012	100707	178063	306310	514381	846627
6	1714	5270	14401	36907	86696	194306	414660	849564	1681744	3230120
7	262	1340	5359	19210	60150	172234	460553	1159167	2774702	6363475
8	7	89	655	3622	16348	63635	223997	720406	2148017	6015901
9	0	1	27	170	1436	8835	44993	196391	762963	2714473
10	0	0	0	0	23	340	3058	20738	114232	547528
11	0	0	0	0	0	1	49	576	5804	42764
12	0	0	0	0	0	0	0	2	56	983
Total	8241	19279	44706	105212	246683	585355	1401644	3381806	8220777	20138200

	M
k	1022	1023	1024	1025	1026	1027	1028	1029	1030
12	10018-
13	55	1277-
14	0	0	1	61	1599	21568
15	0	0	0	2	87	1790
16	0	0	0	0	0	0	51	1642
17	0	0	0	0	0	0	0	1	42

The smallest Carmichael numbers with k prime factors:

k	number	factors
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(10^n-1)

n	C(10n)	C(10n-1)	ratio
4	7	1	7.000
5	16	7	2.286
6	43	16	2.688
7	105	43	2.441
8	255	105	2.429
9	646	255	2.533
10	1547	646	2.395
11	3605	1547	2.330
12	8241	3605	2.286
13	19279	8241	2.339
14	44706	19279	2.319
15	105212	44706	2.353
16	246683	105212	2.345
17	585355	246683	2.373
18	1401644	585355	2.394
19	3381806	1401644	2.413
20	8220777	3381806	2.431
21	20138200	8220777	2.450

The function k(x) of Pomerance/Selfridge/Wagstaff

n	k(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

m	c	25.109	1011	1012	1013	1014	1015
5	0	203	312	627	1330	2773	5814
	1	1652	2785	6575	15755	37467	90167
	2	82	154	327	702	1484	3048
	3	102	172	344	725	1463	3059
	4	124	182	368	767	1519	3124
7	0	401	634	1334	2774	5891	12691
	1	1096	1885	4613	11447	28001	69131
	2	105	186	432	967	2109	4599
	3	152	232	496	1055	2178	4707
	4	129	211	450	985	2122	4592
	5	138	222	454	1033	2224	4777
	6	142	235	462	1018	2181	4715
11	0	335	547	1324	3006	7032	16563
	1	640	1131	2770	6786	16548	40891
	2	139	217	473	1068	2361	5338
	3	142	220	457	1045	2348	5319
	4	104	187	442	1026	2317	5261
	5	152	243	466	1066	2370	5316
	6	116	198	440	1061	2400	5384
	7	122	195	458	1023	2223	5165
	8	129	222	475	1107	2450	5449
	9	131	218	465	1042	2285	5179
	10	153	227	471	1049	2372	5347
12	1	2071	3462	7969	18761	43760	103428
	3	0	0	1	2	2	5
	5	20	32	64	124	228	448
	7	47	75	147	289	547	1027
	9	25	36	60	103	165	294
	11	0	0	0	0	4	10

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

p	25.109	1011	1012	1013	1014	1015	1016	1017
3	25	36	61	105	167	299	565	1025
5	202	309	624	1325	2765	5797	12175	25481
7	364	579	1218	2557	5461	11874	25915	57459
11	263	428	1071	2509	5979	14397	33893	80745
13	237	431	1058	2462	5699	13514	32025	76256
17	117	206	496	1318	3244	8114	20206	50170
19	152	244	532	1401	3358	8141	20020	49413
23	37	78	207	535	1360	3317	8195	20803
29	55	103	284	729	1822	4659	11577	29149
31	101	168	390	876	2116	5153	12575	30667
37	60	95	219	551	1401	3418	8594	21382
41	35	68	171	414	1092	2736	6788	17275
43	35	65	168	403	943	2308	5520	13636
47	14	16	36	81	195	459	1135	2854
53	19	30	55	147	363	973	2327	5842
59	2	4	11	43	100	272	618	1542
61	34	58	148	364	851	1978	4722	11278
67	8	18	50	123	317	815	1950	4843
71	15	25	66	161	389	979	2480	6178
73	14	28	68	175	406	1015	2508	6277
79	4	10	17	66	175	467	1163	2873
83	1	1	4	8	39	79	175	457
89	10	16	23	55	148	409	1003	2523
97	10	20	50	106	261	606	1413	3445

Number of times a prime appears as a factor

p	25.109	1011	1012	1013	1014	1015
3	25	36	61	105	167	299
5	203	312	627	1330	2773	5814
7	401	634	1334	2774	5891	12691
11	335	547	1324	3006	7032	16563
13	483	807	1784	3998	9045	20758
17	293	489	1182	2817	6640	16019
19	372	608	1355	3345	7797	18638
23	113	207	507	1282	3135	7716
29	194	336	832	2094	5158	12721
31	335	571	1320	3086	7270	17382
37	320	535	1270	2926	6826	16220
41	227	390	1001	2418	5896	14344
43	184	296	772	1920	4663	11594
47	53	80	199	492	1223	2873
53	92	160	351	813	2041	5143
59	26	41	92	262	644	1611
61	269	453	1075	2542	6047	14429
67	110	178	407	1063	2540	6306
71	104	194	521	1320	3351	8546
73	198	348	849	2145	4925	11929
79	64	107	247	686	1728	4318
83	14	24	56	137	340	838
89	68	131	320	788	1951	4981
97	123	193	495	1277	3123	7594

The strong Fibonacci pseudoprimes up to 10¹⁷

type	N	factors
LC-	28295303263921	29 31 67 271 331 5237
uLC-	443372888629441	17 31 41 43 89 97 167 331
sLC-	582920080863121	41 53 79 103 239 271 509
LC-	894221105778001	17 23 29 31 79 89 181 1999
LC-	2013745337604001	17 37 41 131 251 571 4159
uLC-	39671149333495681	17 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).

i	N	factors
5	6601	7 23 41
7	561	3 11 17
18	55462177	17 23 83 1709
18	8885251441	11 47 1109 15497
18	42018333841	11 47 1049 77477
21	10585	5 29 73
22	2465	5 17 29
23	1105	5 13 17
25	11921001	3 29 263 521
31	62745	3 5 47 89
37	11972017	43 433 643
37	67902031	43 271 5827
39	334153	19 43 409
43	52633	7 73 103
44	15841	7 31 73
45	8911	7 19 67
47	2821	7 13 31
48	1729	7 13 19
49	1208361237478669	53 653 26479 1318579
50	4199932801	29 499 503 577
52	206955841	17 71 277 619
53	1271325841	17 31 179 13477
54	4169867689	13 29 383 28879
55	271794601	13 19 743 1481
60	6840001	7 17 229 251
61	1962804565	5 103 149 25579
67	410041	41 73 137
67	11985924995083901	29 101 1427 16349 175403
70	162401	17 41 233
76	4752717761	11 17 107 173 1373
81	35575075809505	5 197 223 353 458807
82	1496405933740345	5 47 317 40253 499027
83	142159958924185	5 37 107 58379 123017
83	158664761899885	5 37 107 53987 148469
83	204370370140285	5 37 107 48677 212099
83	24831908105124205	5 29 719 3023 78790717
90	3778118040573702001	11 47 1051 67967 102302009
92	520178982961	11 29 131 607 20507
94	12782849065	5 7 269 317 4283
95	8956911601	11 17 127 131 2879
97	5472940991761	199 241 863 132233
97	721574219707441	167 241 5039 3557977
97	9729822470631481	127 409 110681 1692407
97	83565865434172201	103 1993 9551 42622169
99	438253965870337	43 139 49409 1484009

The Lehmer index of N is l(N) = (N-1)/phi(N).

Carmichael numbers with Lehmer index >= 2.

l(N)	N	factors
2.001626637356340392561983471	3852971941960065	3 5 23 89 113 1409 788129
2.003883612865346524249604640	655510549443465	3 5 23 53 389 2663 34607
2.008372508329582505406681230	13462627333098945	3 5 23 53 197 8009 466649
2.010009765625000000000000000	26708253318968145	3 5 17 113 57839 16025297
2.030673668032786885245901639	269040992399565	3 5 23 29 4637 5799149
2.032073437462429066076752909	158353658932305	3 5 17 89 149 563 83177
2.036135451167959557221302187	1817671359979245	3 5 23 29 359 11027 45893
2.071377818273530507776492047	16057190782234785	3 5 17 29 269 6089 1325663
2.072936866685888594678744261	75131642415974145	3 5 23 29 53 617 9857 23297
2.081287730823863636363636363	881715504450705	3 5 17 47 89 113 503 14543
2.083805744941863020924916476	31454143858820145	3 5 17 23 2129 39293 64109
2.088936015435896686674307775	6128613921672705	3 5 17 23 353 7673 385793
2.098407897763451927081664691	12301576752408945	3 5 23 29 53 113 197 1042133
2.114322378615702479338842975	1886616373665	3 5 17 23 83 353 10979
2.114933090480187426210153482	3193231538989185	3 5 17 23 113 167 2927 9857
2.139359882439079747328472147	11947816523586945	3 5 17 23 89 113 233 617 1409

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