Korselt's criterion can be used the check for them, and they form the only candidates for solutions to Lehmer's Totient problem. In order to be one of these, a number n must divide the denominator of n-1th Bernoulli number, and their general definition requires that n is one if and only if the n-th power-raising function from the ring Z-sub-n to itself is the identity. Alford, Granville, and Pomerance proved in 1994 that there exist infinitely many of these numbers, beginning with 561, 1105, and 1729.  Squarefree and having at least three prime factors, these numbers satisfy the congruence of Fermat's Little Theorem for all choices of base.  
Korselt's criterion can be used the check for them, and they form the only candidates for solutions to Lehmer's Totient problem. In order to be one of these, a number n must divide the denominator of n-1th Bernoulli number, and their general definition requires that n is one if and only if the n-th power-raising function from the ring Z-sub-n to itself is the identity. Alford, Granville, and Pomerance proved in 1994 that there exist infinitely many of these numbers, beginning with 561, 1105, and 1729.  Squarefree and having at least three prime factors, these numbers satisfy the congruence of Fermat's Little Theorem for all choices of base.  For ten points, identify these numbers that are often also called absolute pseudoprimes.
Korselt's criterion can be used the check for them, and they form the only candidates for solutions to Lehmer's Totient problem.
Carmichael numbers