Update on k<300 minmax primes:

Code:

k largest prime
233 179992
13 233207
191 240802
249 251036

All are tested up to n=1M or more.

The largest primes of all other Ks have 300k digits or more.

k=283 was removed from the top of the list by Cruelty.

Its largest prime was at n=149351, but now it's at n=1702599.

The previous status can be found

here.