Ouch, there was a typo in the definition. Here is the message back again

with typo corrected (k-1 changed to k):

It seems the following concept ought to be important:

"k-Generalized Carmichael numbers."

DEFINITION:

If N is composite and squarefree and:

for all primes p that divide N,

p-1 happens to divide N-1,

and:

for each j=2,3,4,...,k,

for all primes p that divide N,

(p^j-1)/(p-1)

happens to divide (N^j-1)/(N-1),

then call N a "k-generalized Carmichael number."

If you make k large enough as a function of N, then I guess

k-generalized Carmichael numbers must stop existing.

QUESTION: how large do you need?

As an initial guess, if k>(logN)*(some positive constant)

that is enough to stop N from being a

k-Generalized Carmichael number.

QUESTION:

Can you find any k-generalized Carmichaels for k=2,3 or 4?

[I believe my computer showed none exist when N<10^16.]