That is a nice sequence. And good thinking about converting it into the k-series (

A258081); one needs a compact representation for a simple sieve and whatnot.

Technically, this series is not

*as fast* as Mersenne's (wall clock wise), because for all terms generic mod reduction on a generic FFT size will be used (making the proof at least twice slower than a similar-sized Mp or a GFN or a cyclotomic GenUni number). My bet for a viz-a-viz Mersenne contender for the largest known prime is between the latter two classes; all they need is a GIMPS' size crowd to follow.

I'll double check your terms with PFGW and scripts. I'll start with writing a quick-and-dirty sieve, maybe tonight.