Since when is prime, all prime numbers are -Lehmer numbers.
Every number which is -Lehmer is also -Lehmer, and thus for simplicity I will call a number -Lehmer only if it is not -Lehmer, and I will consider only composite Lehmer numbers.
The existence of a composite 1-Lehmer number (usually simply called Lehmer number) is still an open problem and several results have been proved about these numbers (which probably do not exist). For example, Cohen and Hagis have proved that such a number, if it exists, must be greater than and be the product of at least 14 primes.
The following table reports the smallest -Lehmer number for from 2 to 36.
Grau & Antonio M. Oller-Marcén have proved several results. For example, that every Carmichael number is also a -Lehmer number.