While looking into the Lucas primality test I noticed an interesting thing. Using the following test* I discovered a sequence of numbers which, for lack of a better name, I'll call the "Poulet-Carmichael numbers":
For all prime factors P of N-1 if the congruence $P^{(N-1)} \equiv 1 \mod N$ holds true then N is either prime or a Poulet-Carmichael number.
The sequence of composites that pass the test includes all of the Carmichael numbers plus a small subset of the super-Poulet numbers. A cursory search of OEIS turned up no result for either that particular sequence nor the non-Carmichael subsequence (12801, 83333, 104653, 113201, 282133, 653333, 721801, 1530787, 1584133, ...).
So my question is simply whether there is any justification whatsoever to classify this sequence as a superset of Carmichaels proper? Without any evidence as such I suppose that would be jumping the gun just a bit. Any thoughts on the matter?
* Incidentally this turns out to be an excellent "pretest" for the Lucas primality test as it tends to filter out all but the most resilient pseudoprimes.