The smaller of the two has now appeared in Top5000 (

http://primes.utm.edu/primes/page.php?id=119934). Interesting things to note:

It has an current rank higher than entry rank (as of this writing)!

It took PFGW 4 passes to prove this (where the N+1 pass should take about 2x N-1 pass)!

BTW, the sister sequence (

https://oeis.org/A258081) is empty? Can someone post all the k values here as a handy reference?

EDIT:- Reconstructed k values from the Top 5000 submission data

Code:

1, 1, 1, 1, 2, 10, 12, 10, 21, 25, 70, 670, 239, 2115, 586, 1619, 26800, 2505, 99019, 40903, 285641, 67166

Has an extra "1" at the beginning to correspond to "2".