The only relation I can find is this:
9998 *10^44-1 is prime
9998*10^4604-1 is prime
9998*10^17780-1 is prime.
According to this, n as been tested up to 230 000 and no other prime was found, so the next term of the sequence (if there is one) must be larger than that.
I thought you already knew that.
I don't think it is possible to calculate the next few terms, but I would gladly be proven wrong.
676754^262144+1 is prime