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Carmichael numbers
Carmichael numbers are composites such that  $a^{n-1}\equiv 1\pmod n$  for every  $1<a<n$  coprime to  $n$, and thus cannot be found to be composite using Fermat's little theorem criterion.

A composite  $n$  is a Carmichael number if and only if it is squarefree and, for every prime  $p$  dividing  $n$,  $p-1$  divides  $n-1$.

Carmichael numbers must be odd and have at least 3 prime factors.

If for a certain  $k$  the 3 numbers  $6k+1$,  $12k+1$  and  $18k+1$  are prime, then their product is Carmichael number.

The first Carmichael numbers are 561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973 more terms

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of Carmichael numbers

A graph displaying how many Carmichael numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of Carmichael numbers
Useful links Mathworld, Carmichael Number
Wikipedia, Carmichael number
OEIS, Sequence A002997
Carmichael numbers can also be... (you may click on names or numbers and on + to get more values)

aban 561 + 175292000065 184030000001 218295000001 530598000241 alternating 561 + 5632705 9494101 41298985 547652161 amenable 561 + 986088961 990893569 993420289 993905641 apocalyptic 1105 + 8911 10585 15841 29341 arithmetic 561 + 9582145 9585541 9613297 9890881 binomial 561 + 716851649251 821743411105 906414656491 921323712961 c.decagonal 2704801 392099401 1030401901 2367379201 369309612001 c.heptagonal 29020321 127664461 c.nonagonal 8911 + 716851649251 821743411105 906414656491 921323712961 c.pentagonal 399001 1746692641 c.square 1105 c.triangular 512461 3858853681 congruent 561 + 8719921 8927101 9494101 9585541 Cunningham 1729 + 110592000001 351596817937 422240040001 432081216001 Curzon 561 2465 151530401 174352641 cyclic 561 + 9582145 9585541 9613297 9890881 d-powerful 2465 278545 3146221 de Polignac 2465 + 75681541 78120001 84350561 92625121 decagonal 1105 + 716726903707 906336844297 944553164101 953828131201 deceptive 1729 + 99751078441 99897989101 99947925121 99976607641 deficient 561 + 9582145 9585541 9613297 9890881 dig.balanced 10585 + 178837201 181154701 184353001 193910977 Duffinian 1105 + 9439201 9494101 9585541 9613297 economical 10585 + 14469841 14676481 15829633 19384289 equidigital 10585 + 14469841 14676481 15829633 19384289 evil 561 + 981567505 985052881 986088961 993905641 Friedman 46657 gapful 561 + 96713938321 97077044701 97492534321 99678195865 happy 62745 + 4463641 4903921 6189121 9439201 Harshad 1729 + 8815102297 9048104209 9216037441 9456330241 heptagonal 670033 173085121 9836283601 hex 8911 + 43331401 511338241 1193229577 50886982081 hexagonal 561 + 716851649251 821743411105 906414656491 921323712961 hoax 656601 27336673 37167361 93614521 Hogben 5310721 2278677961 9593125081 29859667201 iban 41041 101101 410041 interprime 15841 + 43286881 56052361 76595761 88689601 junction 126217 + 67653433 67994641 96895441 99830641 Kaprekar 670033 Lehmer 561 + 998324255809 998667686017 999607982113 999629786233 lucky 1105 + 5310721 6054985 9439201 9890881 magic 1105 2465 magnanimous 2465 6601 modest 340561 nialpdrome 997633 octagonal 2465 + 795949003585 837342821281 870142775041 952910081761 odious 1729 + 977892241 981789337 990893569 993420289 palindromic 101101 pernicious 1729 + 8719309 8830801 8927101 9890881 persistent 17392546081 61280451937 Poulet 561 + 998324255809 998667686017 999607982113 999629786233 Proth 1729 8355841 40280065 53282340865 repunit 5310721 2278677961 9593125081 29859667201 Ruth-Aaron 182356993 2320690177 3203895601 779065788865 self 52633 + 934784929 940123801 955134181 958762729 Smith 656601 27336673 37167361 93614521 sphenic 561 + 90698401 92625121 96895441 99036001 star 258634741 strobogrammatic 101101 super-d 10585 + 6313681 6840001 7995169 9890881 taxicab 1729 triangular 561 + 716851649251 821743411105 906414656491 921323712961 Ulam 41041 + 2628073 4335241 4463641 6733693 unprimeable 449065 + 3224065 3664585 6054985 9582145 wasteful 561 + 9582145 9585541 9613297 9890881 Zeisel 1729 + 600613114501 663805468801 727993807201 856666552249