20210120, 09:43  #1 
Aug 2020
79*6581e4;3*2539e3
3×139 Posts 
2^1571237 · (1153#  1) + 1  why was it constructed like that?
2^1571237 * (1153#  1) + 1 is a new prime submitted to PrimePages.
It is almost a PrimoProth, which are interesting in that they have a low weight, but why the 1? It decreases the number of small factors of N1, but if you're after that, you might just use a prime for k. 
20210120, 12:51  #2 
Aug 2020
79*6581e4;3*2539e3
3×139 Posts 
Correction: Should read "high weight".
And now that I thought about it some more, k = 1153#1 is a large k with few small factors but more elegant than using a 486 digit prime. Maybe that's the reason here. Last fiddled with by bur on 20210120 at 12:52 
20210120, 21:52  #3 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{3}×1,201 Posts 

20210121, 07:36  #4 
Aug 2020
79*6581e4;3*2539e3
3·139 Posts 
I checked other primes of this person and he/she often uses primes or primorials as k. So I assumed it was about having k's with many or few small factors.
But you're right, maybe I'm giving it too much thought. 