--- In

primenumbers@yahoogroups.com, "WarrenS" <warren.wds@...> wrote:

>

> (6 x + 1) (12 x + 1) (18 x + 1)

> is a Carmichael number if all three factors are simultaneously

> prime and where x>0 is integer.

--another remarkable thing about this family by Chernick is that

its "change the sign of 1" version

N = (6 x - 1) (12 x - 1) (18 x - 1)

is another kind of interesting pseudoprime if all three factors are simultaneously prime and where x>0 is integer.

Namely, every prime P dividing N has property that P+1 divides N+1;

this causes certain Lucas-like primality tests to always think N is a prime (unless they find a factor).

----

If N enjoyed both this property and the usual Carmichael property simultaneously, then N would be (in my terminology) a "2-generalized Carmichael."

Broadhurst provided a list by Robert Gerbicz 1 April 2009,

which it turns out was not a joke, of 144153 numbers N.

Also, Broadhurst provided his own list of 246 numbers N.

Unfortunately none of the members of either list

satisfy my criteria for being 2-generalized Carmichael --

that is, I demand that if P is prime and P|N, then (P+1)|(N+1) and (P-1)|(N-1).