A million-digit Leyland prime (encore)

Hot on the heels of my million-digit Leyland prime find, I have come across another one:

197180^119151+1*119151^197180 is 3-PRP!

This one was a little unexpected because it is so close in size to my other one, a mere five decimal digits less. The current top-five Leyland prime leaderboard now looks like this (the first column is the number of digits):

1717671  (1343238,19)      Ryan Propper    May 2023

1433792   (300102,59935)   Ryan Propper    May 2023

1268947  (1139148,13)      Ryan Propper    Jul 2023

1000910   (191319,170462)  Hans Havermann  Jul 2023

1000905   (197180,119151)  Hans Havermann  Aug 2023

833719/265381

Based on the simple continued fraction of π, its convergents (rational approximations) are: 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, ...

Prime numerators are at position 1, 5, 9, ... Prime denominators are at position 2, 4, 9, ... The ninth convergent therefore has both prime numerator and prime denominator, noted ~2003 in the OEIS. It seems unlikely that we will ever see another such.

I thought it might be useful to have here a listing of the positions of prime numerators (p/) and prime denominators (/p) so as to better assess the rarity of their confluence:

1/
/2
/4
5/
9/
/9
11/
16/
/33
87/
230/
334/
594/
/595
840/
853/
/1127
1149/
/2003
2726/
/3611
3788/
/4356
/6926
7442/
8751/
/25333
/27652
/32395
/37722
42038/
/114199
143753/
...