With the new find I did some reading on Thabit/321 primes and read they were originally studied for their relation to amicable numbers. While this is likely known to many here, maybe not everyone knows, so:
If all three of p = 3 * 2^n - 1, q = 3 * 2^(n-1) - 1, and r = 9 * 2^(2n-1) - 1 are prime, then 2^n * p * q and 2^n * r are a pair of amicable numbers.
So far only n = 2, 4, and 7 are known to fulfill the requirements. Are there any ideas as to the density of such numbers n? A very rough estimation of their probability would be (1/ln(p))^3, I guess?
1281979 * 2^485014 + 1 is prime ... no further hits up to: n = 5,600,000