I’ve got you to

Now, let’s keep going. Define and let , , .

Then

Or, alternatively:

In order to satisfy this for all , and . This calls for three relatively prime integers whose product divides their sum. I believe is the only such arrangement, leading us into a dead end.

However, after looking at some raw data, it looks like and subsequent are all over the map. This leads me to pose another conjecture:

Weak: For any three relatively prime , there exists a such that is a three-factor Carmichael Number.

Strong: For any three relatively prime , there exists an such that is a three-factor Carmichael Number iff and is prime for all .

Examples:

For generates Carmichaels 1729, 294409, 56052361, 118901521 …

For generates Carmichaels 29341, 1152271, 64377911, 775368901 …

For generates Carmichaels 2465, 6189121, 19384289, 34153717249 …