A composite n such that an-1 - 1 is divisible by n for each a coprime to n. more
The first 600 Carmichael numbers :
561,
1105,
1729,
2465,
2821,
6601,
8911,
10585,
15841,
29341,
41041,
46657,
52633,
62745,
63973,
75361,
101101,
115921,
126217,
162401,
172081,
188461,
252601,
278545,
294409,
314821,
334153,
340561,
399001,
410041,
449065,
488881,
512461,
530881,
552721,
656601,
658801,
670033,
748657,
825265,
838201,
852841,
997633,
1024651,
1033669,
1050985,
1082809,
1152271,
1193221,
1461241,
1569457,
1615681,
1773289,
1857241,
1909001,
2100901,
2113921,
2433601,
2455921,
2508013,
2531845,
2628073,
2704801,
3057601,
3146221,
3224065,
3581761,
3664585,
3828001,
4335241,
4463641,
4767841,
4903921,
4909177,
5031181,
5049001,
5148001,
5310721,
5444489,
5481451,
5632705,
5968873,
6049681,
6054985,
6189121,
6313681,
6733693,
6840001,
6868261,
7207201,
7519441,
7995169,
8134561,
8341201,
8355841,
8719309,
8719921,
8830801,
8927101,
9439201,
9494101,
9582145,
9585541,
9613297,
9890881,
10024561,
10267951,
10402561,
10606681,
10837321,
10877581,
11119105,
11205601,
11921001,
11972017,
12261061,
12262321,
12490201,
12945745,
13187665,
13696033,
13992265,
14469841,
14676481,
14913991,
15247621,
15403285,
15829633,
15888313,
16046641,
16778881,
17098369,
17236801,
17316001,
17586361,
17812081,
18162001,
18307381,
18900973,
19384289,
19683001,
20964961,
21584305,
22665505,
23382529,
25603201,
26280073,
26474581,
26719701,
26921089,
26932081,
27062101,
27336673,
27402481,
28787185,
29020321,
29111881,
31146661,
31405501,
31692805,
32914441,
33302401,
33596641,
34196401,
34657141,
34901461,
35571601,
35703361,
36121345,
36765901,
37167361,
37280881,
37354465,
37964809,
38151361,
38624041,
38637361,
39353665,
40160737,
40280065,
40430401,
40622401,
40917241,
41298985,
41341321,
41471521,
42490801,
43286881,
43331401,
43584481,
43620409,
44238481,
45318561,
45877861,
45890209,
46483633,
47006785,
48321001,
48628801,
49333201,
50201089,
53245921,
53711113,
54767881,
55462177,
56052361,
58489201,
60112885,
60957361,
62756641,
64377991,
64774081,
65037817,
65241793,
67371265,
67653433,
67902031,
67994641,
68154001,
69331969,
70561921,
72108421,
72286501,
74165065,
75151441,
75681541,
75765313,
76595761,
77826001,
78091201,
78120001,
79411201,
79624621,
80282161,
80927821,
81638401,
81926461,
82929001,
83099521,
83966401,
84311569,
84350561,
84417985,
87318001,
88689601,
90698401,
92625121,
93030145,
93614521,
93869665,
94536001,
96895441,
99036001,
99830641,
99861985,
100427041,
101649241,
101957401,
102090781,
104404861,
104569501,
104852881,
105117481,
105309289,
105869401,
106041937,
107714881,
109393201,
109577161,
111291181,
114910489,
115039081,
115542505,
116682721,
118901521,
119327041,
120981601,
121247281,
122785741,
124630273,
127664461,
128697361,
129255841,
129762001,
130032865,
130497361,
132511681,
133205761,
133344793,
133800661,
134809921,
134857801,
135556345,
136625941,
139592101,
139952671,
140241361,
144218341,
145124785,
146843929,
150846961,
151530401,
151813201,
153589801,
153927961,
157731841,
158404141,
158864833,
159492061,
161035057,
161242705,
161913961,
163954561,
167979421,
168659569,
169057801,
169570801,
170947105,
171454321,
171679561,
172290241,
172430401,
172947529,
173085121,
174352641,
175997185,
176659201,
178451857,
178482151,
178837201,
180115489,
181154701,
182356993,
184353001,
186393481,
186782401,
187188001,
188516329,
188689501,
189941761,
193708801,
193910977,
194120389,
194675041,
196358977,
200753281,
206955841,
208969201,
212027401,
213835861,
214850881,
214852609,
216821881,
221884001,
225745345,
226509361,
227752993,
228842209,
230630401,
230996949,
231194965,
237597361,
238244041,
238527745,
241242001,
242641153,
246446929,
247095361,
250200721,
252141121,
255160621,
256828321,
257495641,
258634741,
266003101,
270857521,
271481329,
271794601,
273769921,
274569601,
275283401,
277241401,
278152381,
279377281,
280067761,
280761481,
288120421,
289766701,
289860481,
291848401,
292244833,
292776121,
295643089,
295826581,
296559361,
299736181,
300614161,
301704985,
302751505,
306871201,
311388337,
318266641,
321197185,
321602401,
325546585,
328573477,
329769721,
333065305,
333229141,
334783585,
338740417,
346808881,
348612265,
354938221,
357277921,
357380101,
358940737,
360067201,
362569201,
364590721,
366532321,
366652201,
367804801,
367939585,
368113411,
382304161,
382536001,
390489121,
392099401,
393122521,
393513121,
393716701,
395044651,
395136505,
396262945,
399906001,
403043257,
403317421,
405739681,
413058601,
413138881,
413631505,
416964241,
417241045,
419520241,
426821473,
429553345,
434330401,
434932961,
438359041,
440306461,
440707345,
455106601,
458368201,
461502097,
461854261,
462199681,
471441001,
471905281,
473847121,
477726145,
481239361,
483006889,
484662529,
490099681,
490503601,
492559141,
496050841,
499310197,
503758801,
507726901,
509033161,
510825601,
511338241,
516684961,
517937581,
518117041,
518706721,
527761081,
529782121,
530443201,
532758241,
533860309,
540066241,
542497201,
544101481,
545363281,
545570641,
547652161,
548871961,
549117205,
549333121,
549538081,
551672221,
552894301,
555465601,
556199281,
556450777,
557160241,
557795161,
558570961,
558977761,
561481921,
561777121,
564651361,
568227241,
569332177,
573896881,
577240273,
579606301,
580565233,
590754385,
593234929,
595405201,
597717121,
600892993,
602074585,
602426161,
602593441,
606057985,
609865201,
611397865,
612347905,
612816751,
616463809,
618068881,
620169409,
621101185,
625060801,
625482001,
629692801,
631071001,
633639097,
638959321,
642708001,
652969351,
656187001,
662086041,
672389641,
683032801,
683379841,
684106401,
686059921,
689880801,
697906561,
698548201,
702683101,
703995733,
704934361,
705101761,
707926801,
710382401,
710541481,
711374401,
713588401,
717164449,
721244161,
722923201,
727083001,
739444021,
743404663,
744866305,
745864945,
746706961,
752102401,
759472561,
765245881,
771043201,
775368901,
775866001,
776176261,
781347841,
784966297,
790020001,
790623289,
794937601,
798770161,
804978721,
809702401,
809883361,
814056001,
822531841,
824389441,
824405041,
829678141,
832060801,
833608321,
834244501,
834720601,
836515681,
839275921,
841340521,
842202361,
843704401,
847491361,
849064321,
851703301,
851934601,
852729121,
854197345.
Distribution of the remainders when the numbers in this family are divided by n=2, 3,..., 11. (I took into account 8241 values, from 561 to 999629786233).
n\r | 0 | 1 |
2 | 0 | 8241 | 2 |
3 | 61 | 8116 | 64 | 3 |
4 | 0 | 8093 | 0 | 148 | 4 |
5 | 627 | 6575 | 327 | 344 | 368 | 5 |
6 | 0 | 8116 | 0 | 61 | 0 | 64 | 6 |
7 | 1334 | 4613 | 432 | 496 | 450 | 454 | 462 | 7 |
8 | 0 | 7099 | 0 | 73 | 0 | 994 | 0 | 75 | 8 |
9 | 0 | 6399 | 25 | 28 | 884 | 22 | 33 | 833 | 17 | 9 |
10 | 0 | 6575 | 0 | 344 | 0 | 627 | 0 | 327 | 0 | 368 | 10 |
11 | 1324 | 2770 | 473 | 457 | 442 | 466 | 440 | 458 | 475 | 465 | 471 |
A pictorial representation of the table above
Imagine to divide the members of this family by a number n and compute the remainders. Should they be uniformly distributed, each remainder from 0 to n-1 would be obtained in about (1/n)-th of the cases. This outcome is represented by a white square. Reddish (resp. bluish) squares represent remainders which appear more (resp. less) frequently than 1/n.